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Quantum Electrodynamics Frontiers in Physics A Lecture Note and Reprint Series DAVID PINES, Editor N. Bloembergen NUCLEAR MAGNETIC RELAXATION: A Re- print Volume Geoffrey F. Chew S-MATRIX THEORY OF STRONG INTER- ACTIONS: A Lecture Note and Reprint Volume R. P. Feynman QUANTUM ELECTRODYNAMICS: A Lecture Note and Reprint Volume R. P. Feynman THE THEORY OF FUNDAMENTAL PROC- ESSES: A Lecture Note Volume Hans Frauenf elder THE MOSSBAUER EFFECT: A Collection of Reprints with an Introduction David Pines THE MANY-BODY PROBLEM: A Lecture Note and Reprint Volume L. Van Hove, N. M. Hugenholtz, and L. P. Howland PROB- LEMS IN THE QUANTUM THEORY OF MANY-PARTICLE SYSTEMS Quantum Electrodynamics A Lecture Note and Reprint Volume R. P. FEYNMAN California Institute of Technology Notes corrected by E. R. HUGGINS H. T. YURA California Institute of Technology W. A. BENJAMIN, INC. New York 1961 QUANTUM ELECTRODYNAMICS A Lecture Note and Reprint Volume Copyright © 1961 by W. A. Benjamin, Inc. All rights reserved. Library of Congress Catalog Card Number: 61-18179 Manufactured in the United States of America c -^ W. A. BENJAMIN, INC. 2465 Broadway, New York 25, New York EDITOR'S FOREWORD The problem of communicating in a coherent fashion the recent developments in the most exciting and active fields of physics seems particularly pressing today. The enormous growth in the number of physicists has tended to make the familiar channels of communication considerably less effective. It has become increas- ingly difficult for experts in a given field to keep up with the cur- rent literature; the novice can only be confused. What is needed is both a consistent account of a field and the presentation of a definite "point of view" concerning it. Formal monographs cannot meet such a need in a rapidly developing field, and, perhaps more im- portant, the review article seems to have fallen into disfavor. In- deed, it would seem that the people most actively engaged in devel- oping a given field are the people least likely to write at length about it. "Frontiers in Physics" has been conceived in an effort to im- prove the situation in several ways. First, to take advantage of the fact that the leading physicists today frequently give a series of lectures, a graduate seminar, or a graduate course in their special fields of interest. Such lectures serve to summarize the present status of a rapidly developing field and may well constitute the only coherent account available at the time. Often, notes on lectures ex- ist (prepared by the lecturer himself, by graduate students, or by postdoctoral fellows) and have been distributed in mimeographed form on a limited basis. One of the principal purposes of the "Frontiers in Physics" series is to make such notes available to a wider audience of physicists. vi > EDITOR'S FOREWORD It should be emphasized that lecture notes are necessarily rough and informal, both in style and content, and those in the series will prove no exception. This is as it should be. The point of the series is to offer new, rapid, more informal, and, it is hoped, more effec- tive ways for physicists to teach one another. The point is lost if only elegant notes qualify. A second way to improve communication in very active fields of physics is by the publication of collections of reprints of recent ar- ticles. Such collections are themselves useful to people working in the field. The value of the reprints would, however, seem much en- hanced if the collection would be accompanied by an introduction of moderate length, which would serve to tie the collection together and, necessarily, constitute a brief survey of the present status of the field. Again, it is appropriate that such an introduction be in- formal, in keeping with the active character of the field. A third possibility for the series might be called an informal monograph, to connote the fact that it represents an intermediate step between lecture notes and formal monographs. It would offer the author an opportunity to present his views of a field that has developed to the point at which a summation might prove extraor- dinarily fruitful, but for which a formal monograph might not be feasible or desirable. Fourth, there are the contemporary classics— papers or lectures which constitute a particularly valuable approach to the teaching and learning of physics today. Here one thinks of fields that lie at the heart of much of present-day research, but whose essentials are by now well understood, such as quantum electrodynamics or magnetic resonance. In such fields some of the best pedagogical material is not readily available, either because it consists of pa- pers long out of print or lectures that have never been published. "Frontiers in Physics" is designed to be flexible in editorial format. Authors are encouraged to use as many of the foregoing approaches as seem desirable for the project at hand. The publish- ing format for the series is in keeping with its intentions. Photo- offset printing is used throughout, and the books are paperbound, in order to speed publication and reduce costs. It is hoped that the books will thereby be within the financial reach of graduate students in this country and abroad. Finally, because the series represents something of an experi- ment on the part of the editor and the publisher, suggestions from interested readers as to format, contributors, and contributions will be most welcome. DAVID PINES Urbayia, Illinois August 1961 PREFACE The text material herein constitutes notes on the third of a three -semes- ter course in quantum mechanics given at the California Institute of Tech- nology in 1953. Actually, some questions involving the interaction of light and matter were discussed during the preceding semester. These are also included, as the first six lectures. The relativistic theory begins in the seventh lecture. The aim was to present the main results and calculational procedures of quantum electrodynamics in as simple and straightforward a way as possi- ble. Many of the students working for degrees in experimental physics did not intend to take more advanced graduate courses in theoretical physics. The course was designed with their needs in mind. It was hoped that they would learn how one obtains the various cross sections for photon processes which are so important in the design of high-energy experiments, such as with the synchrotron at Cal Tech. For this reason little attention is given to many aspects of quantum electrodynamics which would be of use for theo- retical physicists tackling the more complicated problems of the interaction of pions and nucleons. That is, the relations among the many different for- mulations of quantum electrodynamics, including operator representations of fields, explicit discussion of properties of the S matrix, etc., are not in- cluded. These were available in a more advanced course in quantum field theory. Nevertheless, this course is complete in itself, in much the way that a course dealing with Newton's laws can be a complete discussion of me- chanics in a physical sense although topics such as least action or Hamilton's equations are omitted. The attempt to teach elementary quantum mechanics and quantum elec- trodynamics together in just one year was an experiment. It was based on the idea that, as new fields of physics are opened up, students must work VII viii ^ PREFACE their way further back, to earlier stages of the educational program. The first two terms were the usual quantum mechanical course using Schiff (McGraw-Hill) as a main reference (omitting Chapters X, XII, XIII, and XIV, relating to quantum electrodynamics). However, in order to ease the tran- sition to the latter part of the course, the theory of propagation and potential scattering was developed in detail in the way outlined in Eqs. 15-3 to 15-5. One other unusual point was made, namely, that the nonrelativistic Pauli equation could be written as on page 6 of the notes. The experiment was unsuccessful. The total material was too much for one year, and much of the material in these notes is now given after a full year graduate course in quantum mechanics. The notes were originally taken by A. R. Hibbs. They have been edited and corrected by H. T. Yura and E. R. Huggins. R. P. FEYNMAN Pasadena, California November 1961 CONTENTS Editor's Foreword v Preface vii Quantum Electrodynamics 1 Interaction of Light with Matter— Quantum Electrodynamics 3 Resume of the Principles and Results of Special Relativity 23 Relativistic Wave Equation 34 Solution of the Dirac Equation for a Free Particle 56 Potential Problems in Quantum Electrodynamics 71 Relativistic Treatment of the Interaction of Particles with Light 91 Interaction of Several Electrons 118 Discussion and Interpretation of Various "Correction" Terms 128 Pauli Principle and the Dirac Equation 162 Summary of Numerical Factors for Transition Probabilities • Phys. Rev., 84, 123 (1951) 165 The Theory of Positrons • Phys. Rev., 76, 749-759 (1949) 167 Space-Time Approach to Quantum Electrodynamics • Phys. Rev., 76, 769-789 (1949) 178 IX ACKNOWLEDGMENTS The publisher wishes to acknowledge the assistance of the American Institute of Physics in the preparation of this volume, specifically their permission to reprint the three articles from the Physical Review. Quantum Electrodynamics Interaction of Light with Matter- Quantum Electrodynamics First Lecture The theory of interaction of light with matter is called quantum electro- dynamics. The subject is made to appear more difficult than it actually is by the very many equivalent methods by which it may be formulated. One of the simplest is that of Fermi. We shall take another starting point by just postulating for the emission or absorption of photons. In this form it is most immediately applicable. DISCUSSION OF FERMI'S METHOD! Suppose all the atoms of the universe are in a box. Classically the box may be treated as having natural modes describable in terms of a distribu- tion of harmonic oscillators with coupling between the oscillators and matter. The transition to quantum electrodynamics involves merely the assump- tion that the oscillators are quantum mechanical instead of classical. They then have energies (n f l/2)hu;, n = 0, 1 ..., with zero-point energy l/2hu;. The box is considered to be full of photons with a distribution of energies nnu;. The interaction of photons with matter causes the number of photons of type n to increase by ±1 (emission or absorption). Waves in a box can be represented as plane standing waves, spherical waves, or plane running waves exp (iK- x). One can say there is an instan- tRevs. Modern Phys., 4, 87 (1932). QUANTUM ELECTRODYNAMICS taneous Coulomb interaction e 2 /r i j between all charges plus transverse waves only. Then the Coulomb forces may be put into the Schrodinger equa- tion directly. Other formal means of expression are Maxwell's equations in Hamiltonian form, field operators, etc. Fermi's technique leads to an infinite self-energy term e 2 /r i{ . It is pos- sible to eliminate this term in suitable coordinate systems but then the trans- verse waves contribute an infinity (interpretation more obscure). This anom- aly was one of the central problems of modern quantum electrodynamics. Second Lecture LAWS OF QUANTUM ELECTRODYNAMICS Without justification at this time the "laws of quantum electrodynamics" will be stated as follows: 1. The amplitude that an atomic system will absorb a photon during the process of transition from one state to another is exactly the same as the amplitude that the same transition will be made under the influence of a po- tential equal to that of a classical electromagnetic wave representing that photon, provided: (a) the classical wave is normalized to represent an en- ergy density equal to fico times the probability per cubic centimeter of find- ing the photon; (b) the real classical wave is split into two complex waves e -iwt anc j e +iwt ) anc [ on iy the e" iwt part is kept; and (c) the potential acts only once in perturbation; that is, only terms to first order in the electro- magnetic field strength should be retained. Replacing the word "absorbed" by "emit" in rule 1 requires only that the wave represented by exp (+iu)t) be kept instead of exp (— icot). 2. The number of states available per cubic centimeter of a given polar- ization is d 3 K/(27r) 3 Note this is exactly the same as the number of normal modes per cubic cen- timeter in classical theory. 3. Photons obey Bose- Einstein statistics. That is, the states of a collec- tion of identical photons must be symmetric (exchange photons, add ampli- tudes). Also the statistical weight of a state of n identical photons is 1 in- stead of the classical n! Thus, in general, a photon may be represented by a solution of the classi- cal Maxwell equations if properly normalized. Although many forms of expression are possible it is most convenient to describe the electromagnetic field in terms of plane waves. A plane wave can always be represented by a vector potential only (scalar potential made zero by suitable gauge transformation). The vector potential representing a real classical wave is taken as INTERACTION OF LIGHT WITH MATTER A = ae cos (cot - K-x) We want the normalization of A to correspond to unit probability per cu- bic centimeter of finding the photon. Therefore the average energy density should be fico. Now E= (l/c)(8A/8t) = (wa/c)e sinM -K-x) and |B| = |E| for a plane wave. Therefore the average energy density is equal to (1/8tt)(|E| 2 + |B| 2 ) = (l/47r)(w 2 a 2 /c 9 ) sin*M - K-x) = (l/87r)(co 2 a 2 /c 2 ) Setting this equal to hco we find that Thus a = V87rhc 2 /w A = V87rhc z /w e cos (cot - K • x) = v / 47rhc/2o) e {exp[-i(ojt - K«x)] + exp[+i(cut - K» x) ]} Hence we take the amplitude that an atomic system will absorb a photon to be V47rHc 2 /2co exp [-i(u;t - K • x)] For emission the vector potential is the same except for a positive exponen- tial. Example: Suppose an atom is in an excited state *j with energy E. and makes a transition to a final state * f with energy E f . The probability of transition per second is the same as the probability of transition under the influence of a vector potential ae exp[+i(cot - K-X)] representing the emit- ted photon. According to the laws of quantum mechanics (Fermi's golden rule) Trans, prob./sec = 2tt/H | f (potential) . | 2 • (density of states) ., , K 2 dK dfi w 2 dn Density of states = (27r) J d(coh) (27rcrfi QUANTUM ELECTRODYNAMICS The matrix element U fi = | f (potential)^ 2 is to be computed from pertur- bation theory. This is explained in more detail in the next lecture. First, however, we shall note that more than one choice for the potential may give the same physical results. (This is to justify the possibility of always choos- ing = for our photon.) Third Lecture The representation of the plane-wave photon by the potentials A(x, t) = ae exp [-i(ojt - K • x)] 0=0 is essentially a choice of "gauge." The fact that a freedom of choice exists results from the invariance of the Pauli equation to the quantum-mechanical gauge transform. The quantum-mechanical transformation is a simple extension of the classical, where, if •V + 90/ at and B = Vx A and if x is an Y scalar, then the substitutions A = A +Vx 0' = + 9x/9t leave E and B invariant. In quantum mechanics the additional transformation of the wave function is introduced. The invariance of the Pauli equation is shown as follows. The Pauli equation is n d* _ 1 i at 2m (p-±Af..(,-iAJ ^ + e0^ Then, since INTERACTION OF LIGHT WITH MATTER dx dx dx dx Vie' 1 **) =e- J Mp-Vx)* and The partial derivative with respect to time introduces a term (ax/3t)*e" ix , and this may be included with 0e" ix *. Therefore the sub- stitutions *' - e iX^ A' = A e _ + -v x 0' = + Ox/9t) leave the Pauli equation unchanged. The vector potential A as defined for a photon enters the Pauli Hamil- tonian as a perturbation potential for a transition from state i to state f. Any time -dependent perturbation which can be written AH = e iwt U(x,y,z) results in the matrix element U fi given by U fi = /^♦AH^jd vol = f<p{*(x) exp[i(E f /R)t] e iwt U(x) exp[-i(E f /h)t] ^(x)d vol This expression indicates that the perturbation has the same effect as a time- independent perturbation U(x,y,z) between initial and final states whose en- ergies are, respectively, E.- w * and E f . As is well knownf the most impor- tant contribution will come from the states such that E f = E. -coh. Using the previous results, the probability of a transition per second is -r^ ^^ 27T |TT ,0 00 2 dfi P fi d = — | Uft | * w fSee, for example, L. D. Landau and E. M. Lifshitz, "Quantum Mechan- ics; Non-Relativistic Theory,'' Addison-Wesley, Reading, Massachu- setts, 1958, Sec. 40. QUANTUM ELECTRODYNAMICS To determine U fi , write H = T L fp- -A)' - ^-(g.VxA) +eV 2m \ c / 2mc e 2 2mc 2 A- A Because of the rule that the potential acts only once, which is the same as requiring only first-order terms to enter, the term in A • A does not en- ter this problem. Making use of A = ae exp [— i(ajt — K • x)] and the two operator relations (1) Vx A = Kxe e + iK ' x e lu,t (2) pe +iK ' x = e +iK * x (p -HK) or p. ee +iK'X =e +iK« *(p. e -RKe) where K-e = (which follows from the choice of gauge and the Maxwell equations), we may write u fi = aj(/) f *[-(e/2mc)(p-e e + iK * x + e +iK * x e-p) + (ehi/2mc) a • (K x e) e + iK * x ] <p { d vol This result is exact. It can be simplified by using the so-called "dipole" approximation. To derive this approximation consider the term (e/2mc)(p-e e +lK " x ), which is the order of the velocity of an electron in the atom, or the current. The exponent can be expanded. e+ iK- x = 1 + iK . x + !/ 2 (iK- x) 2 + •• • K- x is of the order a /A., where a = dimension of the atom and A = wave- length. If a /A« 1, all terms of higher order than the first in a A ma Y De neglected. To complete the dipole approximation, it is also necessary to neglect the last term. This is easily done since the last term may be taken as the order of (RK/mc) = (hKc/mc 2 ) « (mv 2 /2mc 2 ). Although such a term is negligible even this is an overestimate. More correctly, INTERACTION OF LIGHT WITH MATTER (eRi/2mc)o- • (K x e) e +iK ' x « v/c x [matrix element of (J • (KX p)] The matrix element is J <f> f *<r • (Kxp)0 i d vol A good approximation allows the separation f * = f *(x)U f *(spin) and 0i = 0i (x)U i *(spin) Then to the accuracy of this approximation the integral is JVf*<x)0 1 (x)U f *(o-- (Kxp))Ui d vol = since the states are orthogonal. For the present, the dipole approximation is to be used. Then e Pfi' e U fi = -a 11 cm where Pfi • e = /0f*(P-e)tf>i =e • J0 f *P0i d vol So 2 p «-ffe']*---> ,d ° Cc (2tt) 3 Using operator algebra, p fi /m = haj fi x f . , so that P fl dfi = a 2 [e 2 ^ 4 /(27r) 2 ](e- x fi ) 2 dfi where x fi = J f * x 0j d vol. The total probability is obtained by inte- grating P fi over dQ, thus 2 i. Total prob./sec = J a 2 ^~J (© " x fi ) 2 dfi = a 2 ^ i /|x fi | 2 S in 3 «d9 Z7T = a 2 4e 2 co 4 |x f1 | 2 /67r 10 QUANTUM ELECTRODYNAMICS The term e • x fi is resolved by noting (Fig. 3-1) |x fi • e| = |x f} | sin G FIG. 3-1 Substituting for a' Total prob./sec = - |- ^j |x fi | 2 Fourth Lecture Absorption of Light. The amplitude to go from state k to state 1 in time T (Fig. 4-1) is given from perturbation theory by a lk = -U/h) Jo expQ-E,t)u lk (t) exp^-^Ektjdt FIG. 4-1 INTERACTION OF LIGHT WITH MATTER 11 where the time dependence of U kl (t) is indicated by writing U lk (t)=u, k e- lwt (In accord with the rules of Lecture 2, the argument of the exponential is minus and only terms which are linear in the potential are included.) Using this time dependence and performing the integration, [i< E >- exp — (Ej-nw-Ek) - 1 a lk = El -hou-E k Ulk the transition probability is given by 2 _ 4sin 2 (AT/2n) , |a ]k | - A 2 l u ik I A-E 1 -E k -Bo; This is the probability that a photon of frequency w traveling in direction (6, 0) will be absorbed. The dependence on the photon direction is contained in the matrix element u lk . For example, see Eq. (4-1) for the directional dependence in the dipole approximation. If the incident radiation contains a range of frequencies and directions, that is, suppose (^probability that a photon is present with fre- P(ct),0,0)da; df2 = J quency oj to cu + dco and in solid angle dfi [.about the direction (6,cp) and the probability of absorption of any photon traveling in the (6,<p) direc- tion is desired, it is necessary to integrate over all frequencies. This ab- sorption probability is r°° 4 sin 2 (AT/2h) , l2 ™ n v , n „ J A 2 " ' Ulkl p < w ' ^) dw dn when T is large, the factor (A)" 2 sin 2 (AT/2h) has an appreciable value only for Hco near Ej-E k , and ~P(<jo,6,(p) will be substantially constant over the small range in w which contributes to the integral so that it may be taken out of the integral. Similarly for u lk , so that Trans, prob. = 2 7 r(rl)- 1 |u lk | 2 P(w lk ,0,0)dfi (4-1) where ha; lk = (Ej-E k ) 12 ^ QUANTUM ELECTRODYNAMICS This can also be written in terms of the incident intensity (energy crossing a unit area in unit time) by noting that Intensity = i(oj,9,(p)du dQ = hwc P(o>,0,0) dco dfi Thus Trans, prob. - 27r(h)- 1 |u lk | 2 (fico^c)- 1 i(co lk ,0,0)dfi (4-2) Using the dipole approximation, in which u ]k = V27T/W (e/mc)(p lk • e) = V27r/co lk (e/c)ho; lk (x lk • e) the total probability of absorption (per second) is 47r 2 e 2 (hc)- 1 (x lk • e) 2 i(w lk , 6, 0) dti (4-3) It is evident that there is a relation between the probability of spontane- ous emission, with accompanying atomic transition from state 1 to state k, Probability of spontaneous 1 = , , , , dfi emission/sec J '" K11 1K and the absorption of a photon with accompanying atomic transition from state k to state 1, Eq. (4-1), although the initial and final states are re- versed since |u lk | = |u kl | . This relation may be stated most simply in terms of the concept of the probability n(oj,^,0) that a particular photon state is occupied. Since there are (27rc)T 3 <jo 2 du) dQ photon states in frequency range dco and solid angle d£2, the probability that there is some photon within this range is P(w,0,0) doo dfi = n(co,0,0)(2coc)~ 3 oj 2 da) dfi Expressing the probability of absorption in terms of n(o; > 0,0), Trans, prob. /sec = 27r(h) _1 |u lk | 2 n(a),0,0)(27rc)- 3 u; kl 2 d£3 (4-4) This equation may be interpreted as follows. Since n(a;, #,</>) is the prob- ability that a photon state is occupied, the remainder of the terms of the right-hand side must be the probability per second that a photon in that state will be absorbed. Comparing Eq. (4-4) with the rate of spontaneous emis- sion shows that INTERACTION OF LIGHT WITH MATTER 13 'Prob./sec of absorption "1 of a photon from a state >■ .(per photon in that state) J prob./sec of spontaneous emission of a photon into that state In what follows, it will be shown that Eq. (4-4) is correct even when there is a possibility of more than one photon per state provided n(co,0,0) is taken as the mean number of photons per state. If the initial state consists of two photons in the same photon state, it will not be possible to distinguish them and the statistical weight of the initial state will be 1/2! However, the amplitude for absorption will be twice that for one photon. Taking the statistical weight times the square of the ampli- tude for this process, the transition probability per second is found to be twice that for only one photon per photon state. When there are three pho- tons per initial photon state and one is absorbed, the following six processes (shown on Fig. 4-2) can occur. Any of the three incident photons may be absorbed and, in addition, there is the possibility that the photons which are not absorbed may be interchanged. The statistical weight of the initial state is 1/3 !, the statistical weight of the final state is 1/2! , and the amplitude for the process is 6. Thus the transi- tion probability is (1/3 !)(l/2!)(6) 2 = 3 times that if there were one photon per initial state. In general, the transition probability for n photons per initial photon state is n times that for a single photon per photon state, so Eq. (4-4) is correct if n(ou, B,(p) is taken as the mean number of photons per state. 14 QUANTUM ELECTRODYNAMICS A transition that results in the emission of a photon may be induced by incident radiation. Such a process (involving one incident photon) could be indicated diagrammatically, as in Fig. 4-3. FIG. 4-3 One photon is incident on the atom and two indistinguishable photons come off. The statistical weight of the final state is 1/2 ! and the amplitude for the process is 2, so the probability of emission for this process is twice that of spontaneous emission. For n incident photons the statistical weight of the initial state is 1/n!, the statistical weight of the final state is l/(n + 1) !, and the amplitude for the process is (n + 1) ! times the amplitude for spontaneous emission. The probability (per second) of emission is then n + 1 times the probability of spontaneous emission. The n can be said to account for the induced part of the transition rate, while the 1 is the spon- taneous part of the transition rate. Since the potentials used in computing the transition probability have been normalized to one photon per cubic centimeter and the transition prob- ability depends on the square of the amplitude of the potential, it is clear that when there are n photons per photon state the correct transition prob- ability for absorption would be obtained by normalizing the potentials to n photons per cubic centimeter (amplitude yfn times as large). This is the basis for the validity of the so-called semiclassical theory of radiation. In that theory absorption is calculated as resulting from the perturbation by a potential normalized to the actual energy in the field, that is, to energy nfiu; if there are n photons. The correct transition probability for emission is not obtained this way, however, because it is proportional to n + 1. The er- ror corresponds to omitting the spontaneous part of the, transition prob- ability. In the semiclassical theory of radiation, the spontaneous part of the emission probability is arrived at by general arguments, including the fact that its inclusion leads to the observed Planck distribution formula. Ein- stein first deduced these relationships by semiclassical reasoning. INTERACTION OF LIGHT WITH MATTER 15 Fifth Lecture Selection Rules in the Dipole Approximation. In the dipole approximation the appropriate matrix element is x if = J * f *x*j d vol The components of jaff x if are x if> y if , z if and Trans, prob. « |x if | 2 + |y if | 2 + |z if | 2 Selection rules are determined by the conditions that cause this matrix ele- ment to vanish. For example, if in hydrogen the initial and final states are S states (spherically symmetrical), X if = and transitions between these states are "forbidden." For transitions from P to S states, however, x if £ and they are "allowed." In general, for single electron transitions, the selection rule is AL = ±1 This may be seen from the fact that the coordinates x, y, and z are essen- tially the Legendre polynomial Pj. If the orbital angular momentum of the initial state is n, the wave function contains P n . But P t P n = [l/(2n+ 1)] [nP n _ t + (n+ l)P n+1 ] Hence for the matrix element not to vanish, the angular momentum of the final state must be n ± 1, so that its wave function will contain either P n+ i or P n _i. For a complex atom (more than one electron), the Hamiltonian is H =£)(l/2m)[P a - ( e /c)A(x a )] 2 + Coulomb terms a The transition probability is proportional to |P mn | 2 = |2(P a ) mn | 2 , where the sum is over all the electrons of the atom. As has been shown, (P a ) mn is the same, up to a constant, as (x a ) mn , and the transition probability is propor- tional to l X mn| 2 = £< X «>n a In particular, for two electrons the matrix element is /# f *(x 1 ,x 2 )(x 1 + x 2 )* i (x 1 ,x 2 ) dxj dx 2 Xj + x 2 behaves under rotation of coordinates similarly to the wave function of some "object" with unit angular momentum. If the "object" and the atom 16 QUANTUM ELECTRODYNAMICS in the initial state do not interact, then the product (x t + ^2)^1 (Xi>x 2 ) can be formally regarded as the wave function of a system (atom + object) having possible values of Jj + 1, Jj , and J. — 1 for total angular momentum. There- fore the matrix element is nonzero only if J f , the final angular momentum, has one of the three values J i ± 1 or J } . Hence the general selection rule AJ =±1, 0. Parity. Parity is the property of a wave function referring to its behavior upon reflection of all coordinates. That is, if *(-Xi, -x 2 , ....) = +*(Xi,X 2 , ...) parity is even; or if *(-Xi, -x 2 , ...) = -*(xi,x 2 , ...) parity is odd. If in the matrix elements involved in the dipole approximation one makes the change of variable of integration x = -x', the result is x if = /*f *(x)x*j(x) d 3 x = J* f *(-x , )(-x , )*o(-x / ) dV If the parity of * f is the same as that of *j, it follows that x if = -x if = Hence the rule that parity must change in allowed transitions. For a one- electron atom, L determines the parity; therefore, AL = would be forbid- den. In many-electron atoms, L does not determine the parity (determined by algebraic, not vector, sum of individual electron angular momenta), so AL = transitions can occur. The 0— transitions are always forbidden, however, since a photon always carries one unit of angular momentum. All wave functions have either even or odd parity. This can be seen from the fact that the Hamiltonian (in the absence of an external magnetic field) is invariant under the parity operation. Then, if H^(x) = E*(x), it is also true that H*(-x) = E\&(-x). Therefore, if the state is nondegenerate, it follows that either *(-x) = *(x) or *(-x) = -^(x). If the state is degen- erate, it is possible that \fr(-x) * *(x). But then a complete solution would be one of the linear combinations *(x) + ^(-x) even parity ^(x) - *(-x) odd parity Forbidden Lines. Forbidden spectral lines may appear in gases if they are sufficiently rarefied. That is, forbiddenness is not absolute in all cases. It may simply mean that the lifetime of the state is much longer than if it INTERACTION OF LIGHT WITH MATTER 17 were allowed, but not infinite. Thus, if the collision rate is small enough (collisions of the second kind ordinarily cause de-excitation in forbidden cases), the forbidden transition may have sufficient time to occur. In the nearly exact matrix element f* f *(e- p)e~ iK ' x *j d 3 x the dipole approximation replaces e" iK * x by 1. If this vanishes, the transi- tion is forbidden, as described in the foregoing. The next higher or quadru- pole approximation would then be to replace e" lK * x by 1 - i/K- x, giving the matrix element -i /# f *(e -p)(K- x)*id 3 x For light moving in the z direction and polarized in the x direction, this becomes -iK /* f *(p x z) *.d 3 x - -iK| f (p x z)j and the transition probability is proportional to whereas in the dipole approximation it was proportional to lf(Px)i| 2 Therefore the transition probability in the quadrupole approximation is at least of the order of (Ka) 2 = a 2 /ft , smaller than in the dipole approxima- tion, where a is of the order of the size of the atom, and ft the wavelength emitted. Problem: Show that H(xz) - (xz)H = (h/mi)(p x z + xp z ) and consequently that [(K/mi)(p x z + xp z )] mn = (xz) mn (E m -E n ) Note that p x z can be written as the sum p x z = 1/2 (p x z + xp z ) + l/2(p x z - xp z ) From the preceding problem, the first part of p x z is seen to be equivalent, up to a constant, to xz, which behaves similarly to a wave function for angu- QUANTUM ELECTRODYNAMICS lar momentum 2, even parity. The second part is the angular momentum operator L , which behaves like a wave function for angular momentum 1, even parity. Therefore the selection rules corresponding to the first part are seen to be A J = ±2, +1, with no parity change. This type of radiation is called electric quadrupole. The selection rules for the second part of p x z are A J = ±1, 0, no parity change, and the corresponding radiation is called magnetic dipole. Note that unless A J ±2, the two types of radiation cannot be distinguished by the change in angular momentum or parity. If A J = ±1, 0, they can only be distinguished by the polarization of the radiation. Both types may occur simultaneously, producing interference. In the case of electric quadrupole radiation, it is implicit in the rules that 1/2 — 1/2 and 0—1 transitions are forbidden (even though A J may be ± 1), since the required change of 2 for the vector angular momentum is im- possible in these cases. Continuing to higher approximations, it is possible by similar reasoning to deduce the vector change in angular momentum, or angular momentum of the photon, and the selection rules for parity change and change of total an- gular momentum A J associated with the various multipole orders (Table 5-1). TABLE 5-1. Classification of Transitions and Their Selection Rules Electric Magnetic Electric Magnetic Electric Multipole dipole dipole quadrupole quadrupole octupole ^1 Angular 1 1 2 2 3 A ► momentum fy J Parity Odd Even Even Odd Odd Parity change Yes No No Yes Yes Change of total ±1,0 ±1,0 ±2,±1,0 ±2, ±1,0 ±3,±2,±1,0| I angular mo- mentum A J No 0—0 No 0—0 No 0-0 1 1 2 2 No 0-0 1 1 2 2 No 0-0 1 1 2 2 ^ 0—1 0—1 etc. (see i following) 1 j Actually all the implicit selection rules for A J, which become numerous for the higher multipole orders, can be expressed explicitly by writing the selection rule as |J f -J, 1 < J, + J where 2 is the multipole order or 1 is the vector change in angular mo- mentum. INTERACTION OF LIGHT WITH MATTER 19 It turns out that in so-called parity-favored transitions, wherein the prod- uct of the initial and final parities is (— l)Jf " ^i and the lowest possible mul- tipole order is J f - Jj , the transition probabilities for multipole types con- tained within the dashed vertical lines in Table 5-1 are roughly equal. t In parity-unfavored transitions, where the parity product is (— lyf~h +1 and the lowest multipole order is | J f — Jj | + 1, this may not be true. Sixth Lecture Equilibrium of Radiation. If a system is in equilibrium, the relative num- ber of atoms per cubic centimeter in two states, say 1 and k, is given by Nj/N k = e- (E l- E kV kT =e - hw / kT according to statistical mechanics, when the energies differ by ho;. Since the system is in equilibrium, the number of atoms going from state k to 1 per unit time by absorption of photons ftco must equal the number going from 1 to k by emission. If n w photons of frequency u> are present per cubic cen- timeter, then probabilities of absorption are proportional to n w and proba- bility of emission is proportional to n + 1. Thus N k n w =N 1 (n a; +l) (n- w + l)/n w = N k /N, =e*u/ kT n w =l/(e*"^ T -1) This is the Planck black-body distribution law. The Scattering of Light. We discuss here the phenomena of an incident photon being scattered by an atom into a new direction (and possibly energy) (see Fig. 6-1). This may be considered as the absorption of the incoming photon and the emission of a new photon by the atom. The two photons taking part in the phenomenon are represented by the vector potentials. A t -(27r/o; 1 ) 1 / 2 e 1 e +i(w l t - K * x) A 2 = (27r/co 2 ) 1 / 2 e 2 e- i(w 2t-K-x) The number to be determined is the probability that an atom initially in state k will be left in state 1 by the action of the perturbation A = Aj + A 2 in the or t For nuclei emitting gamma rays this does not seem to be true. For an obscure reason the magnetic radiation predominates for each order of mul- tipole. 20 QUANTUM ELECTRODYNAMICS No. 2 FIG. 6-1 time T. This probability can be computed just as any transition probability with the use of A lk , where A lk = 6 kl expf-KEj/fiyr] - (i/h) x j Q exp[-i(E 1 /H)(T-t 3 )] U lk (t 3 ) exp[-i(E k /h)t 3 ] dt 3 + S / J t4 exp[-i(Ei/n)(T-t 4 )] xU ln (t 4 )exp[-i(E n /h)(t 4 -t 3 )] U nk (t 3 ) exp[-i(E k /h)t 3 ] dt 3 dt 4 The dipole approximation is to be employed and U = AH = (e/2mc)(p*A) + (e 2 /mc 2 )(A'A) where spins are neglected. In each integral defining A lk , each of the two vector potentials must ap- pear once and only once. Thus, in the first integral the term p • A of U will not appear in U lk . The product A • A = (A { + A 2 ) • (A t + A 2 ) will contribute only its cross-product term 2AiA 2 . The second integral will have no con- tribution from A- A , but will be the sum of two terms. The first term con- tains a U ln based on p • A 2 and a U nk based on p- A t . The second has U ln based on p • Aj and U nk on p • A 2 . The time sequences resulting in these two terms can be represented schematically as shown in Fig. 6-2. The integral resulting from the first term will now be developed in de- tail. (P'Ai) nk =(2V^i) 1 ^ 2 (p-e 1 ) nk er lw i* (p-A 2 )i n = (27r/cD 2 ) 1/2 (p-e 2 ) ln e W2 t Then the resulting integral is INTERACTION OF LIGHT WITH MATTER 21 No. 1 No. 2 atom No. 1 FIG. 6-2 ^27r/(a) 1 cc; 2 ) 1 /^p.e 2 ) ln (p.e 1 ) nk n. x / / 4 exp[-i(E,/H)(T-t 4 ) + iw 2 t 4 ] x expf-KE^^^-taJ-iw^] exp[-i(E k /K)t 3 ] dt 3 dt 4 The integral is similar to the integrals considered previously with regard to transition probabilities, and the sum becomes £ 27r/(a; 1 a; 2 ) 1 / 2 (p • e^ (p • e 2 ) nk e 1 * n x [sin (T • A/fi)/(E k - E n + Kc^) -A] where A = (E 1 +ho) 2 - E k -ft(jo{), and the phase angle <p is independent of n. A term with the denominator given by (E n -Rc^- E k )(Ej +ho; 2 - E n ) has been neglected, since previous results show that only energies such that Ej + hco 2 a E k + hwi are important. The final result can be written Trans, prob./sec = (2tt/R)|M| 2 [co 2 2 dft 2 /(27r 3 )] (6-1) (JC where |M| is determined from A lk by integrating over co 2 and averaging over e 2 . Then the complete expression for the cross section cr is cr dfi 5 2 4 a »'2 _l_y> (P-e2)i n (P-ei)nk mV E k +ho; 1 -E n (p- ei ) ln (p>e 2 ) nk i E k -E n -Rc 2 + ^<°i" e »>fiik (6-2) 22 QUANTUM ELECTRODYNAMICS The first term under the summation comes from the "first term" pre- viously referred to and the second from the "second term." The last term in the absolute brackets comes from A- A. If 1 * k, the scattering is incoherent, and the result is called the "Raman effect." If 1 = k, the scattering is coherent. Further, note that if all the atoms are in the ground state and 1 *k, then the energy of the atom can only increase and the frequency of the light w can only decrease. This gives rise to "Stokes lines." The opposite effect gives "anti-Stokes lines." Suppose wi = co 2 (coherent scattering) but further Kw t is very nearly equal to E k -E n , where E n is some possible energy level of the atom. Then one term in the sum over n becomes extremely large and dominates the remain- der. The result is called "resonance scattering." If a is plotted against w, then at such values of w the cross section has a sharp maximum (see Fig. 6-3). FIG. 6-3 The "index" of refraction of a gas can be obtained by our scattering for- mula. It can be obtained, as for other types of scattering, by considering the light scattered in the forward direction. Self-Energy. Another phenomenon that must be considered in quantum electrodynamics is the possibility of an atom emitting a photon and reabsorb- ing the same photon. This affects the diagonal element A kk . Its effect is equivalent to a shift of energy of the level. One finds (P-e) kn (p-e) nk d 3 K 2tt e = e/ E k -E n -u> (27rh) 3 co where e is the direction of polarization. This integral diverges. A more exact relativistic calculation also gives a divergent integral. This means that our formulation of electromagnetic effects is not really a completely satisfactory theory. The modifications required to avoid this difficulty of the infinite self-energy will be discussed later. The net result is a very small shift AE in position of energy levels. This shift has been observed by Lamb and Rutherford. Resume of the Principles and Results of Special Relativity Seventh Lecture The principle of relativity is the principle that all physical phenomena would appear to be exactly the same if all the objects concerned were mov- ing uniformly together at velocity v; that is, no experiments made entirely inside of a closed spaceship moving uniformly at velocity v (relative to the center of gravity of the matter in the universe, for example) can determine this velocity. The principle has been verified experimentally. Newton's laws satisfy this principle; for they are unchanged when subject to a Galli- lean transformation, x' = x — vt y' = y z' = z t' = t because they involve only second derivatives. The Maxwell equations are changed, however, when subjected to this transformation, and early workers in this field attempted to make an absolute determination of velocity of the earth using this feature (Michelson-Morley experiment). Failure to detect any effects of this type ultimately led to Einstein's postulate that the Max- well equations are of the same form in any coordinate system; and, in par- ticular, that the velocity of light is the same in all coordinate systems. The transformation between coordinate systems which leaves the Maxwell equa- tions invariant is the Lorentz transformation: x cosh u - ct sinh u Vl-(v 2 /c 2 ) 23 24 QUANTUM ELECTRODYNAMICS y =y t -(xv/c 2 ) X . , _/, . ■ - — sinh u + t cosh u /l - (v 2 /c 2 ) c where tanh u = v/c. Henceforth we shall use time units so that the speed of light c is unity. The latter form is written to demonstrate the analogy with rotation of axes, x 7 = x cos 9 + y sin 9 y' = —x sin 9 + y cos 9 Successive transformations v t and v 2 or u A and u 2 add in the sense that a single transformation v 3 or u 3 will give the same final system if v 3 - v'i + v 2 or tanh u 3 = tanh (u t + u 2 ) Einstein postulated (theory of special relativity) that the Newton laws must be modified in such a way that they, too, are unchanged in form under a Lorentz transformation. An interesting consequence of the Lorentz transformation is that clocks appear to run slower in moving systems; that is called time dilation. In transforming from one coordinate system to another it is convenient to use tensor analysis. To this end, a four-vector will be defined as a set of four quantities that transforms in the same way as x,y,z and ct. The subscript \i will be used to designate which of the four components is being considered; for example, Xj = x x 2 = y x 3 = z x 4 = t The following quantities are four-vectors: a d d d ■ . — — ~ , ~ t~ "j > + Z~ (V) four- dimensional gradient dx dy dz at l V & j x > j y . j z > P (jju) current (and charge) density A x , A y , A z , cp (A ) vector (and scalar) potential P x » Py P z » ^ (pj momentum and total energy t 2 |The energy E, here, is the total energy including the rest energy mc . SPECIAL RELATIVITY 25 An invariant is a quantity that does not change under a Lorentz transforma- tion. If a and b are two four-vectors, the "product" a ' b -S a fi b M ~ a 4 b 4 " a l b l " a 2 b 2 - a 3 b 3 n is an invariant. To avoid writing the summation symbol, the following sum- mation convention will be used. When the same index occurs twice, sum over it, placing minus in front of first, second, and third components. The Lorentz invariance of the continuity equation is easily demonstrated by writ- ing it as a "product" of four-vectors V and j : d P aj x aj y aj z v M j M =v 4 j 4 -y 1J1 -v 2 j 2 -v 3 J3=- + — + ^r+^ Conservation of charge in all systems if it is conserved in one system is a consequence of the invariance of this "product," the four-dimensional di- vergence V- j. Another invariant is P ll P u =P-P = E 2 - p x 2 - p y 2 - p z 2 = E 2 - p 2 = m 2 (E = total energy, m = rest mass, mc 2 = rest energy, p = momentum.) Thus, E 2 = p 2 c 2 + m 2 c 4 It is also interesting to note that the phase of a free particle wave function exp [(-i/h)(Et-p-X)] is invariant since Et - p-X = Et - p x x - py - p z z =p p /j^M The invariance of p^p^ can be used to facilitate converting laboratory en- ergies to center-of-mass energies (Fig. 6-4) in the following way (consider identical particles, for simplicity): E iab E E Or 9 c^ ^^ moving stationary particle particle Laboratory system Center-of-mass system FIG. 6-4 26 QUANTUM ELECTRODYNAMICS PmP u = E ubm = E +p ' U^U but P<> = E, m so E lab m = 2E 2 - m 2 and E o = [im(E lab + m)] 1/2 The equations of electrodynamics B = V X A and E = - (l/c)(a A/at) - V are easily written in tensor notation, B x =8A z /8y - d\/dz = -V y A z +V z A y B y - 3A x /az - dA z /dx = - V Z A X +V X A Z B z ^aAy/ax- 9A x /ay = -V x A y +V y A x E x = - 9A x /9t - a (p/dx = -v t A x + V x A t E y = -8A /8t - 80/ay = -V t A y + V y A t aA z /at - a^/az •V t A z + V z A t where use is made of the fact that is the fourth component of the four- vector potential A . From the foregoing it can be seen that B x , B y , B z , E x , E y , and E z are the components of a second-rank tensor: F = V A - V A (7-D This tensor is antisymmetric (F = -F y ) and the diagonal terms (jj. - v) are zero; thus there are only six independent components (three components of E and three components of B) instead of sixteen. "B z B Y E B 7 "B x E F P, - " B Y B x E "Ex - E Y -E z The Maxwell equations Vx B = 4?r J + (9E/8t) and V-E = 4irp are written V M F u, =4^ (7-2) SPECIAL RELATIVITY 27 where v = 1, 2, 3, 4, that is, jj = j x , j 2 = j y , J3 = j z , J4 = p, and /i is a dummy index of summation. The v - 1, 2, and 3 gives the three components of the curl equation, and v - 4 gives the divergence equation. The equation satisfied by the potential A^ is found, by substituting Eq. (7-1) into Eq. (7-2), to be VVA — V V A = 47ri The potential A y is not unique, however, since the potential A/ M =A M +V M X ( 7 "3) (X = any scalar function of position) also satisfies this relation. Such a change or transformation of potential is called a gauge transformation (for historical reasons). We shall make the potentials more definite by assum- ing that all potentials have been transformed so as to satisfy the so-called Lorentz condition! V M A M =0 (7-4) This is convenient, because it simplifies the equation for A„ to (V-V)A y =47rj y (7-5) since V-V = ^n^u » which can be recognized as the wave equations V 2 A - a 2 A/9t 2 - -4ttj (7-5') V 2 0- d 2 (p/dt 2 = -47rp Sometimes Eq. (7-5') is written n 2 A M = -47rj M (D 2 = D'Alembertian opera- tor = V 2 - O/at) 2 = -V-V). This choice of gauge (V^ A^ = 0) is the usual one made in classical electrodynamics, V- A -30/at = (7-4') t This is not sufficient to completely define A. We may still use any x such that D 2 x = 0. 28 ^ QUANTUM ELECTRODYNAMICS Eighth Lecture SOLUTION OF THE MAXWELL EQUATION IN EMPTY SPACE In empty space the plane wave solution of the wave equation □ 2 A M =-4ttj m =0 is where e and k are constant vectors, and k„ is subject to the condition that k^k^ =k-k =0 This may be seen from the fact that V v operating on e" lk ' x has the effect of multiplying by ik y (V y does not operate on e. since the coordinates are rectangular). Thus, -D 2 A, =V>„A,)=V„<-ie M k 1 ,e- il '; x ) = -^(k„ye- ik ' X Note that in these operations V y A actually forms a second-rank tensor, V y (V y A ) a third-rank tensor, and then contraction on the index v yields a first-rank tensor or vector. The k„ is the propagation vector with components k^ = co, K x , K y , K z = co, K so that in ordinary notation exp (-ik • x) = exp [-i (cot - K • X)] and the condition k • k = means co 2 - K-K = Problem: Show that the Lorentz condition V p A M = implies that k • e =0. When working in three dimensions it is customary to take the polariza- tion vector e such that K -e = and to let the scalar potential 0=0. But SPECIAL RELATIVITY 29 this is not a unique condition; that is, it is not relativistically invariant and will be true only in a one-coordinate system. This would seem to be a para- dox attaching some uniqueness to the system in which K • e = 0, a situation incompatible with relativity theory. The "paradox," however, is resolved by the fact that one can always make a so-called gauge transformation, which leaves the field F^ unaltered but which does change e. Therefore, choosing K-e = in a particular system amounts to selecting the certain gauge. The gauge transformation, Eq. (7-3), is A' = A + V X 0' =0 +Ox/9t) where x is a scalar. But V • A = 0, the Lorentz condition, Eq. (7-4), will still hold if V • A' =V-A + V-x =0 or if □ 2 x =0 This equation has a solution x = ae*" ik ' X , so A M = A m + V a e_lk ' X ) =(e p +ak M )e- ik '* where a is an arbitrary constant. Therefore, is the new polarization vector obtained by gauge transformation. In ordinary notation e' =e + aK ej = e 4 + aoo Thus, no matter what coordinate system is used, K-e' = K • e + a K • K = K • e + aoo 2 can be made to vanish by choice of the constant a . Clearly the field is left unchanged by a gauge transformation for F W = v „ K ~ \ A '» = V M A„ + V,V„ X - \A V - V„V p x = F„„ 30 QUANTUM ELECTRODYNAMICS the V^V^X = ^V^x because the order of differentiations is immaterial. RELATIVISTIC PARTICLE MECHANICS The components of ordinary velocity do not transform in such a manner that they can be components of a four-vector. But another quantity dz/ds = dt/ds, dx/ds, dy/ds, dz/ds where dz = dt, dx, dy, dz is an element of path of the particle and ds is the proper time defined by ds 2 = dt 2 - dx 2 - dy 2 - dz 2 is a four- vector and is called the four-velocity u . Dividing ds 2 by dt 2 gives the relation between proper time and local time to be (ds/dt) 2 = 1 - v 2 The components of ordinary velocity are related as follows: dx/ds = (dx/dt) (dt/ds) = v x /(l -y 2 ) 1/2 dy/ds =v y /(l-v 2 ) 1 / 2 dz/ds =v z /(l-v 2 ) 1 / 2 dt/ds = 1/(1 -v 2 ) 1 / 2 It is evident that u„u„ = 1, for 1 v x 2 v y 2 v z 2 1 - v 2 u m u m = tt^2 - yzt^ - i _ v 2 " T-V = T^7 = 1 The four-momentum is defined P M =mu M =m/(l- v 2 ) 1/2 , mv x /(l-v 2 ) 1/2 , mv y /(l - v 2 ) 1/2 , mv z /(l - v 2 ) 1/2 Note that p 4 = m/(l - v 2 ) 1/2 is the total energy E, so that in ordinary nota- tion the momentum P is given by SPECIAL RELATIVITY 31 P = Ev where v is the ordinary velocity. Like the velocity, the components of ordinary force defined by d/dt (mo- mentum) cannot form the components of a four-vector. But the quantity f„=dP M /ds does form a four-vector with the components f u = d/dt(mv M /VI "-' v 2 ) dt/ds = F^/Jl - v2 M = 1, 2, 3 where F is the ordinary force. The fourth component is power _ rate of change of energy _ d/dt(m/Vl - v 2 ) 4 ~ vr 1- ^ " vi - v 2 vi - v2 _ This is seen from the fact that m/Vl - v 2 is the total energy and also from the ordinary identity Power = F • V _d mV 1 dt VI - v 2 J ni v 2 1 dv 2 2 (l-v 2 ) 3/2 + (1-v 2 ) 1 / 2 dt mv dv ( l_ v 2)3/2 dt dt jj_ yl Thus the relativistic analogue of the Newton equations is d/ds (p^) = f M = m d 2 z M /ds 2 (8-1) The ordinary Lorentz force is F = e(E + v X B) (8-2) and the rate of change of energy is F-v = eE-v Then from the preceding definition of four-force, f = e/(l - v 2 ) 1/2 (E + vxB) and 32 QUANTUM ELECTRODYNAMICS f 4 = e/(l - v 2 ) 1/2 E-v Problem: Show that the expressions just given for f and f 4 are equivalent to f = eu F so that the relativistic analogue of the Newton equation becomes m d 2 z p /ds 2 = e(dz y /ds) F^ (8-3) Also show that this implies d/ds[(dz^/ds) 2 ] =0 In ordinary terms the equation of motion is d/dt(mv/Vl - v 2 ~) = e(E + v X B) (8-4) It can be shown by direct application of the Lagrange equations d/dtOL/3v J[i )-OL/ax M ) = that the Lagrangian L = -m/l - v 2 - e0+ eA- v (8-5) leads to these equations of motion. Also the momenta conjugate to x is given by 9 L/8v or P =mv/(l - v 2 ) 1/2 + eA The corresponding Hamiltonian is H = e<p + [(P - eA) 2 + m 2 ] 1/2 (8-6) which satisfies (H-e0) 2 - (P-eA) 2 = m 2 . It is difficult to convert the Hamiltonian idea to a covariant or four-dimensional formulation. But the principle of least action, which states that the action S = j L dt shall be a minimum, will lead to the relativistic form of the equations of motion directly when expressed as SPECIAL RELATIVITY 33 S = Jl dt = m /ds + e J A^ (dz p /ds) ds = J[m(dz/da • dz/da) 1/2 + eA^ dz^/da] da Note that by definition (ds/da) 2 = {dz ll /da){dz u /da) It is interesting that another "action, " defined S' =m/2 j (dz^/da) 2 da + e j A jl (z jl )(dz jI /da) da leads to the same result as for S in the foregoing. Problems: (1) Show that the Lagrangian, Eq. (8-5), leads to the equations of motion, Eq. (8-4), and that the corresponding Hamiltonian is Eq. (8-6). Also find the expression for P. (2) Show that 6S = (va- riation of S), where S is the action just given, leads to the same equa- tions . Relativistic Wave Equation Ninth Lecture UNITS The following convention will be used hereafter. We define the units of mass and time and length such that c = 1 (c = 2.99 793 x 10 10 cm/sec) R = 1 (h = 1.0544 x 10~ 27 erg/sec) Table 9-1 (top of page 35) is given as a useful reference for conversion to customary units. The following numerical values are useful: M p = mass of proton - 1836.1 m = 938.2 Mev Mass unit of atomic weights = 931.2 Mev M H = Mass of hydrogen atom = 1.00815 mass units M N = Mass of neutron = 784 kev + M H kT = 1 ev when T = 11,606° K N a = Avogadro's number = 6.025 x 10 23 N.e = 96,520 coulombs KLEIN-GORDON, PAULI, AND DIRAC EQUATIONS According to relativistic classical mechanics, the Hamiltonian is given by H = V(p - eA) 2 + m 2 + e0 (9-1) 34 RELATIVISTIC WAVE EQUATION 35 TABLE 9-1. Notations and Units Present Customary notation Meaning notation Value m Mass of electron m Energy mc 2 510.99 kev Momentum mc 1704 gauss cm Frequency mc 2 /K Wave number mc/fi 1/m Length (Comp- ton wave- length)/27r fi/mc 3.8615 x 10 -11 cm Time H/mc 2 e 2 Fine -structure constant (dimensionless) e 2 /hc 1/137.038 e 2 /m Classical radius of the electron e 2 /mc 2 2.8176 x lO'^cm 1/me 2 Bohr radius a = H 2 /me 2 0.52945 A If the quantum -mechanical operator -iV is used for p, the operation deter- mined by the square root is undefined. Thus the relativistic quantum- mechanical Hamiltonian has not been obtained directly from the classical equation, Eq. (9-1). However, it is possible to define the square of the oper- ator and to write (H -e0) 2 - (p- eA) 2 = m 2 Then, if H = i8/9t, [-(fi/i)8/8t - e0] 2 *-[(R/i)(a/ax) - e/cA x ] 2 * - ... = m 2 * (9-2) where the square of an operator is evaluated by ordinary operator algebra. This equation was first discovered by Schrodinger as a possible relativistic equation. It is usually referred to as the Klein-Gordon equation. In relativ- istic notation it is (iV^ - eA M )(iV M - eA^)* = m 2 * (9-2') 36 QUANTUM ELECTRODYNAMICS This equation does not allow for "spin" and therefore fails to describe the fine structure of the hydrogen spectrum. It is proposed now for applica- tion to the it meson, a particle with no spin. To demonstrate its application to the hydrogen atom, let A = and = -Ze/r, then let * = x (r) exp (-iEt). Then the equation is (E - Ze 2 /r) 2 x + V 2 X = m 2 X Let E = m + W, where W« m, and substituting V = Ze 2 /r, (W - V)x + V 2 x /2m = -(W - V) 2 x/2m Neglecting the term on the right in comparison with the first term on the left gives the ordinary Schrodinger equation. By using (W - V) 2 /2m as a perturbation potential, the student should obtain the fine-structure splitting for hydrogen and compare with the correct values. Exercise; For the Klein-Gordon equation, let p = i(ty*dty/dt - * 8 * */ 9 1) — e0*** = charge density j = — i(^*V^ — #V^*) — eA^^* = current density Then show (p, j) is a four-vector and show V j = 0. The Klein-Gordon equation leads to a result that seemed so unreasonable at the time it was first brought to light that it was considered a valid basis for rejecting the equation. This result is the possibility of negative energy states. To see that the Klein-Gordon equation predicts such energy states, consider the equation for a free particle, which can be written D 2 ^ = m 2 * where D 2 is the D'Alembertian operator. In four-vector notation, this equa- tion has the solution V = A exp(-ip p x M ), where p^Py = m 2 . Then, since P^P M =P 4 P4 ~ PxPx" PyPy " PzPz = E 2 - p • p there results E =±(m 2 + p-p) 1/2 The apparent impossibility of negative values of E led Dirac to the de- velopment of a new relativistic wave equation. The Dirac equation proves to be correct in predicting the energy levels of the hydrogen atom and is the accepted description of the electron. However, contrary to Dirac's original RELATIVISTIC WAVE EQUATION 37 intent, his equation also leads to the existence of negative energy levels, which by now have been satisfactorily interpreted. Those of the Klein- Gordon equation can also be interpreted. Exercise: Show if ^ = exp(-iEt)x (x,y,z) is a solution of the Klein- Gordon equation with constant A and 0, then ^ = exp(+iEt)x * is a so- lution with -A and -<fi replacing A and 0. This indicates one manner in which "negative" energy solutions can be interpreted. It is the solution for a particle of opposite charge to the electron, but the same mass. Instead of following the original method in the development of the Dirac equation, a different approach will be used here. The Klein-Gordon equation is actually the four- vector form of the Schrodinger equation. With an anal- ogous point of view, the Dirac equation can be developed as the four-vector form of the Pauli equation. In following such a procedure, the terms involving "spin" will be included in the relativistic equation. The idea of spin was first introduced by Pauli, but it was not at first clear why the magnetic moment of the electron had to be taken as he/2mc. This value did seem to follow naturally from the Dirac equation, and it is often stated that only the Dirac equation produces as a consequence the correct value of the electron's magnetic moment. However, this is not true, as further work on the Pauli equation showed that the same value follows just as naturally, i.e., as the value that produces the greatest simplification. Because spin is present in the Dirac equation, and absent in the Klein-Gordon, and because the Klein-Gordon equation was thought to be invalid, it is often stated that spin is a relativistic requirement. This is in- correct, since the Klein-Gordon equation is a valid relativistic equation for particles without spin. Thus the Schrodinger equation is H# = E* where H = l/2m(-iV- eA) 2 + e0 and the Klein-Gordon equation is [(H - e<p) 2 - (-iV - eA) 2 ] * = m 2 * (9-3) Now the Pauli equation is also H\fr = E\£, where H = (l/2m)[a ■ (-iV- eA)] 2 + e0 (9-4) Thus ( -iV — eA) 2 appearing in the Schrodinger equation has been replaced by [o - • (— iV— eA)] 2 . Then a possible relativistic version of the Pauli equa- tion, in analogy to the Klein-Gordon equation, might be 38 QUANTUM ELECTRODYNAMICS (H - e0) 2 * -{a- [(fi/i)V- (e/c)A]} 2 * = m 2 * Actually, this is incorrect, but a very similar form [with H replaced by i(9/at)] is correct, namely, [iO/8t) - e0 - <r • (-iV- eA)] x [iO/at) - e0 + <r • (-i V- eA)] * = m 2 * (9-5) This is one form of the Dirac equation. The wave function ^ on which the operations are being carried out is actually a matrix: •■-© A form closer to that originally proposed by Dirac may be obtained as follows. For convenience, write I(8/8't) - e0 = tt 4 -iV - (e/c)A = ir Now let the function x be defined by (ir i + cr • ir) * = mx . Then Eq. (9-5) implies (7r 4 - a • tt)x = m*. This pair of equations can be rewritten (only to arrive at a particular conventional form) by writing X + * = * a X - * = % Then adding and subtracting the pair of equations for *,x , there results -7r 4 * b + q • 7T ^ a = m* b (9-6) These two equations may be written as one by employing a particular convention. Define a new matrix wave function as * = T a 2 (9-7) where the matrix character of * a and * b has been shown explicitly, i.e. actually I RELATIVISTIC WAVE EQUATION 39 •-(fc) «-(fc) Then, if the auxiliary definitions are made, 74 = {Note: An example of the latter definition is / 1* I 1 ° 1 \ o o I v° ° \0 - "J -1/ I I <7 (9-8) 0i V 1 1 -1 since ^x = V 1 o y y and y z are similar.) The two equations in \fr a and * b can be written as one in the form y 4 7r 4 ^ - y • 7r* = m* which is actually four equations in four wave functions. Then using four- vector notation, the Dirac equation is y^^ = m* or y. M (iV M - eA u )* =m* (9-9) Exercise: Show rnTy + r y r, if /x * y 2 if /* = i; = 4 -2 if v =M = 1, 2, 3 that is, show Vt =1 VtTx =i -YxV t y x 2 = y y 2 = y z 2 = -i 'x'y ^vYx etc. 40 QUANTUM ELECTRODYNAMICS A similar form for the Dirac equation might be obtained by a different argument, by comparison to the Klein-Gordon equation. Thus with H = i(9/9t) = i V 4 and with ecp = eA 4 , Eq. (9-3) becomes (iV M - eA^) 2 * = m 2 # (9-10) in four-vector notation. Using a similar notation in the Pauli equation, Eq. (9-4), but also using a = y and setting a 4 = y 4 arbitrarily (to complete the definition of a four-vector form of a), Eq. (9-4) can be written in a form similar to Eq. (9-10), {y M l(R/i)V^ - (e/c)A^]} 2 M/ = m 2 * (9-11) This should be compared to Eq. (9-9). Now the Pauli equation, Eq. (9-4), differs from the Schrodinger equation in the replacement of the three-dimensional scalar product (p - eA) 2 by the square of a single quantity cr • (p - eA). Analogously one might guess that the four-vector product (p^ — eA ) 2 in Eq. (9-10) must be replaced by the square of a single quantity y (p — eA ), where we must invent four ma- trices y„ in four dimensions in analogy to the three matrices a in three dimensions. The resulting equation, [y M (iV M -eA^)] 2 * =m 2 * (9-11) is essentially equivalent to Eq. (9-9) (operate on both sides of Eq. (9-9) by y M (iV^ - eA^) and use Eq. (9-9) again to simplify the right-hand side). Exercise: Show that Eq. (9-11) is equivalent to (iV M - eA p ) 2 - 1 ey^F^ * = m 2 * Tenth Lecture ALGEBRA OF THE y MATRICES In the preceding lecture the Dirac equation, y M (iV M - eA M )* =m* (10-1) was obtained, together with a special representation for the y's, RELATIVISTIC WAVE EQUATION 41 where each element in these four-by-four matrices is another two-by-two matrix, that is, 1 = (o 1) unit matrix °* = (1 o) etc ' The best way to define the y 's, however, is to give their commutation re- lationships, since this is all that is important in their use. The commutation relationships do not determine a unique representation for the y 's, and the foregoing is only one of many possible representations. The commutation relationships are VtTx.y, z + rx,y, Z Tt =0 ( 10 " 3 ) TxTy + TyTx = TxTz + T Z Tx = ° TyTz + TzTy = or, in a unified notation, 6..,. =0 fji * v = +1 \x = v = 4 = -1 // = v = 1,2,3 Note that with this definition of 6„ y and the rule for forming a scalar prod- uct, Other new matrices may arise by forming products of the matrices al- ready defined. For example, the matrices of Eq. (10-5) are products of y's taken two at a time. The matrices T x r y T x y z y y y z y x y t y y y t y z y t are all independent of y x , y y , y 7 , y t . (They cannot be formed by a linear combination of the latter.) Similarly, products of three matrices, y x y y y z (= y^y t ) y y y z y t (=- y x ys) y z y t y x (= - y y ys) y t y x y y (=- y z ys) 42 QUANTUM ELECTRODYNAMICS These are the only new products of three. For, if two of the matrices were equal, the product could be reduced, thus y t y y y t = — TtTt^y = - Yy The only new product of four that can be formed is given a special name, y 5 , Products of more than four must contain two equal so that they can be re- duced. There are, therefore, sixteen linearly independent quantities. Linear combinations of them may involve sixteen arbitrary constants. This agrees with the fact that such a combination can be expressed by a four-by-four ma- trix. (It is mathematically interesting then that all four-by-four matrices can be expressed in the algebra of the y 's; this is called a Clifford algebra or hypercomplex algebra. A simpler example is that of two-by-two matrices, the so-called algebra of quaternions, which is the algebra of the Pauli spin matrices.) Exercise: Verify that iTxTy (?.y -V* -c-y »■'.- (?:>« and that TtYx,y,z = ( n X ' y,Z ) ~ a ( defmition of &) \ °x y z / y It is convenient to define another y matrix, since it occurs frequently: (10-6) 75 = 3 ') 1 0/ Verify that Wt <: -;) Wx,y,z -( T5 2 = - 1 Wn + y M T5 = o For later use, it will be convenient to define * = a ^ M - a t"Kt " a xYx - a yTy - a zTz from which it can be shown that = -#£ + 2a-b (a-b = a M b p ) a 2 - a , a p K,y,z ) (10-7) (10-8) *y 5 = -y 5 * (io-9) RELATIVISTIC WAVE EQUATION 43 For example, the first may be verified by writing = (a t y t -a x y x -a y y y -a z y z )(b t y t -b x y x -b y y y -b z y z ) and, moving the second factor to the front, by using the commutation rela- tionships. Doing this with the first term, (b t y t ) of the second factor produces b tVt( a tVt + a xT x + a y y y + a 2 y z ) since y t commutes with itself and anticommutes with y x , y y , and y z . By performing this operation on all terms, one obtains = b t y t [(-a t y t + a x y x + a y y y + a z y z ) + 2a t y t ] + b xrx[(a t 7t - a xT x - VV - a z y z ) + 2a xy x ] + b y y y [(a t y t - a x y x - a y y y - a z y z ) + 2a y y y ] + b z y z [(a t y t - a x y x - a^ - a z y z ) + 2a z y z ] = -W + 2(b t a t y t 2 + b x a x y x 2 + b y a y y y 2 + b z a z y z 2 ) = -W + 2b • a Exercises: (1) Show that y x ay x = j£ + 2a x y x y^u =4 T/i^Tju =4a-b y M *Wy M =-2eW (2) Verify by expanding in power series that exp [(u/2)y t y x ] = cosh (u/2) + y t y x sinh (u/2) exp[(0/2)y x y y ] = cos (0/2) + y x y y sin (6/2) (10-10) (3) Show that exp[-(u/2)y t y z ]y t exp [+(u/2)y t y z ] = y t cosh u + y z sinh u 44 QUANTUM ELECTRODYNAMICS exp[-(u/2)y t y z ] y 2 exp [+(u/2)y t y z ] = y 2 cosh u + y t sinh u exp[-(u/2)y t y 2 ]y y exp[+ (u/2)y t y 2 ] = y y exp[-(u/2)y t y 2 ]y x exp [+(u/2)y t y 2 ] = y x (10-11) EQUIVALENCE TRANSFORMATION Suppose another representation for the y's is obtained which satisfies the same commutation relationships, Eq. (10-3); will the form of the Dirac equa- tion, Eq. (10-1), remain the same? To answer this question, make the fol- lowing transformation of the wave function * = S^F, where S is a constant matrix which is assumed to have an inverse S -1 (SS" 1 = 1). The Dirac equa- tion becomes y p 7r M S^ = mS*' (10-12) The 7r and S commute, since it is a differential operator plus a function of position, so this equation may be written y S7r M *' =mS*' Multiplying by the inverse matrix, S-iy Sir *' = mS~ 1 S^' or y ' 7r M ^' = m*' where y' = S _1 y S. The transformation y' - S _1 y S is called an equiva- lence transformation, and it is easily verified that the new y 's satisfy the commutation relationships, Eq. (10-3). Products of y 's, y'^v =(S- 1 y,S)(S~ 1 y y S)^S- 1 (y M y y )S transform in exactly the same manner as the y 's, so that equations involv- ing the y 's (the commutation relations specifically) are the same in the transform representation. This demonstrates another representation for the y's, and the Dirac equation is in exactly the same form as the original, Eq. (10-1), and is equivalent in all its results. RELATIVISTIC INVARIANCE The relativistic invariance of the Dirac equation may be demonstrated by assuming, for the moment, that y transforms similarly to a four-vector. RELATIVISTIC WAVE EQUATION 45 That is, "Kx = (Tx - vy t )/(l - v 2 ) 1/2 7t = (T t " vy x )/(l - v 2 ) 1/2 Jy = Ty 72 = Tz Also 7r transforms similarly to a four-vector because it is a combination of two four- vectors V and A . The left-hand side y tt of the Dirac equa- tion is the product of two four-vectors and hence invariant under Lorentz transformations. The right-hand side m is also invariant. Transforming y u as a four-vector means a new representation for the y 's, but Eqs. (10-11) can be used to show that the new y 's differ from the old y 's by an equiva- lence transformation; thus it is really not necessary to transform the y 's at all. That is, the same special representation can be used in all Lorentz coordinate systems. This leads to two possibilities in making Lorentz transformations: 1. Transform the y 's similarly to a four-vector and the wave function remains the same (except for Lorentz transformation of coordinates). 2. Use the standard representation in the Lorentz -transformed coordinate system, in which case the wave function will differ from that in (1) by an equivalence transformation. HAMILTONIAN FORM OF THE DIRAC EQUATION To show that the Dirac equation reduces to the Schrodinger equation for low velocities, it is convenient to write it in Hamiltonian form. The original term, Eq. (10-1), may be written yj-(fi/i) (3/9t) - e0]* - y • [(H/i)V-eA]* = m* Multiplying by cy t and rearranging terms gives -(fi/i)0*/at) = {y t y • [(fi/i)V - eA] + e0 + y t m}* = H* By Eq. (10-5), H is written H = a • [(K/i) V - eA] + ecf) + m/3 where /3 = y t , a X) y )Z = ytVx.y.z' Ec l- ( 10_5 )» and tne a ' s satisfy the follow- ing commutation relations: a x 2 = a y 2 = a z 2 =/3 2 = 1 and all pairs anticommute. It will be noted that a,/3 are Hermitian matrices in our special represen- tation, so that in this representation H is Hermitian. 46 QUANTUM ELECTRODYNAMICS Exercise: Show that a probability density p = &*$? and a probability current j = **a^ satisfy the continuity equation Op/at) + V- j =0 Note: * is a four- component wave function and p = **# = ($f*f*f*f)[ ^ 2 ) = tff*! + *2**2 + *3** 3 + *** *4 Eleventh Lecture It should be noted that p and a are Hermitian only in certain representa- tions. In particular, they are Hermitian in the representation employed thus far; this will be called the standard representation and expressions in it will be labeled S.R. when appropriate. The Hermitian property of a and £ is necessary in order to get p = \j/*\£ j =**a* S.R. (11-1) as the expressions for charge and current density. Hence they are not true in all representations. The Dirac equation is (with R, c restored) -(K/i)0¥/8t) = H* H =/3mc 2 + e0 + ca • [(fi/i) V - (e/c)A] (ll-2)t t It is noted that the Hamiltonian found in Schiff (' 'Quantum Mechanics," McGraw-Hill, New York, 1949) differs from this one by negative signs on all but the e<p term. Also the components * lf \£ 2 , ^3, ^4 of the wave function used in Schiff correspond, respectively, to -* b , -^ b , -* a , * a here. All this is the result of an equivalence transformation S 2 = i^a x cx y c^ z between the representations used here and in Schiff. It is easily verified that S 2 = -1 hence S _1 = -S and s-i 1 1 1 -1 RELATIVISTIC WAVE EQUATION 47 The expected value of x is <x> = J **x^ d vol = /Ofrfx*! + **X* 2 + *3*X* 3 + * 4 *X* 4 ) d Vol S.R. remembering that # now is a four-component wave function. Similarly it may be verified as an exercise that <oj> = J y?*aV d vol < a x> = J(*4**i + ^3**2 t ***3 + ^1**4) d vol S.R. Also matrix elements are formally the same as before. For example, <«)mn =/*J«*n d Vo1 If A is any operator then its time derivative is A =i(HA -AH) + 8 A/at For X the result is clearly x = i (Hx -xH) = a (11-3) since x commutes with all terms in H except p-oj. But a 2 = 1, so the eigenvalues of a are ±1. Hence the eigenvelocities of x are ± speed of light. This result is sometimes made plausible by the argument that a pre- cise determination of velocity implies precise determinations of position at two times. Then, by the uncertainty principle, the momentum is completely uncertain and all values are equally likely. With the relativistic relation be- tween velocity and momentum, this is seen to imply that velocities near the speed of light are more probable, so that in the limit the expected value of the velocity is the speed of light. t Similarly, (p - eA) x =i(Hp x - p x H) - ie(HA x - A^) - e9A x /at = -e(8 0/8x) + ea*(3A/8x)-e(a-V)A x -e(8A x /at) The terms in A and A x , except the last, expand as follows: f This argument is not completely acceptable, for X commutes with p; that is, one should be able to measure the two quantities simultaneously. 48 , QUANTUM ELECTRODYNAMICS dA x dA v dA z dA x d\ x 3x y 3x 3x x 8x y oy 2 dz J This seen to be the x component of e a x (V x A) = e a x B The first and last terms form the x component of E. Therefore, (p - eA) =e(E + a X B) = F where F is the analogue of the Lorentz force. This equation is sometimes regarded as the analogue of Newton's equations. But, since there is no di- rect connection between this equation and x , it does not lead directly to Newton's equations in the limit of small velocities and hence is not com- pletely acceptable as a suitable analogue. The following relations may be verified as true but their meaning is not yet completely understood, if at all: (d/dt) [x + (i/2m)/3a] = 03/m)(p - eA) (d/dt) [t + (i/2m)/3] - (/3/m)(H - e0) i(d/dt)(a x a y a z ) = -2mPa x a y a z -(d/dt)(j8a) =2((3a x a y a z )(v- eA) where in the last relation <r means the matrix a ,0 <7 so that cr z = -ia x a y , etc. From analogy to classical physics, one might expect that the angular mo- mentum operator is now L = Rx (p - eA) Note that in classical physics p - eA = mv (1 - v 2 r 1/2 From previous results for R and (p - eA), the time derivative of L may be written RELATIVISTIC WAVE EQUATION 49 L = Rx(p-eA) +Rx(p-eA) = a x (p - eA) + R x F The last term may be interpreted as torque. For a central force F, this term vanishes. But then it is seen that L * because of the first term; that is, the angular momentum L is not conserved, even with central forces. But consider the time derivative of the operator <7 defined as <J or where cr z = -a x a y , etc. The z component is seen to commute with the p, e<p, and a 2 terms of H but not with the a x and a y terms, so that cr z = + l(Ha x a y - a x a y H) = + (a x ir x a x a y - a x a y a x n x + a y n y a x a y - OL x a y ci y -n y ) t where 7r = (-iV - eA) But a x n x a x a y = a x oi x a y Tr x = oyr x -oi x a y a x 7r x = a x a x a y ir x = a y it x a y ir y a x a y = -a y a y a x ir y = -a x n y -a x a y a y ir y = -a x n y so that cr z - (2a y ?r x - 2a x ir y ) This is seen to be the z component of -2a x 7r. Finally then, 1/2 (<r) = - a x ir = -a x(p- eA) and this is the first term of L with negative sign. Therefore it follows that (d/dt)[L+ (H/2)cr] = Rx F which vanishes with central forces. The operator L + (R/2)o" may be re- garded as the total angular momentum operator, where L represents orbi- tal angular momentum and (h/2)<7 intrinsic angular momentum for spin 1/2. Thus total angular momentum is conserved with central forces. 50 QUANTUM ELECTRODYNAMICS Problems: (1) In a stationary field 0=0, 3A/at = 0, show that a • (p - eA) is a constant of the motion. Note that this is a consequence of the anomalous gyromagnetic ratio of the electron. It also means that the cyclotron frequency of the electron equals its rate of precession in a magnetic field. (2) In a stationary magnetic field 0=0, 3A/8t = 0, and for a sta- tionary state, show that ^ lf ^ 2 in are the same as ty t , ^ 2 in the Pauli equation. Also, if E Pauli is the kinetic energy in the Pauli equation and E Dirac - W + m is the rest plus kinetic energy in the Dirac equation, show that E Dirac = ^ 2 ™ E Pauli + ^ and explain the simplicity of this relationship. NONRELATIVISTIC APPROXIMATION TO THE DIRAC EQUATION It will be assumed that all potentials are stationary and stationary states will be considered. This makes the work simpler but is not necessary. In this case * =e~ iEt *(X) H* = E^ (Dirac Hamiltonian) and put That is, E =m + W H* = (m + W)* = a - (p - eA) ^ + p m* + e0^ It will be recalled with ^ written as Eq. (9-5) and with a, ft as given in Lecture 10, the previous equation may be written as two equations (9-4'), (m + W)* a = a- 7r* b + m* a + V* a (11-4) RELATIVISTIC WAVE EQUATION 51 (m + W)* b = o- • Trt? a - m* b + V* b (11-5) where, as before, tt = (p - eA) and V = e0. Simplifying and solving Eq. (11-5) for * b gives * b = [l/(2m + W - V)] (a . 7r)* a (11-6) It is noted that if W and V are « 2m, then # b ~ (v/c)* a . For this reason ^ a and ^ b are sometimes referred to as the large and small components of ^, respectively. Substitution of * b from Eq. (11-6) into Eq. (11-4) gives W* a = (a-7r)[l/(2m+W-V)](<r.7r)* a + V* a (11-7) and, if W and V are neglected in comparison to 2m, the result is W* a = (l/2m)(<r-7r) 2 * a + V* a This is the Pauli equation, Eq. (9-4). Now the approximation will be carried out to second order, that is, to order v 2 /c 2 , to determine just what error may be expected from use of the Pauli equation. Twelfth Lecture Using the results of Lecture 11, given by Eqs. (11-6) and (11-7), the low- energy approximation (w - V) « 2m will be made, keeping terms to order v 4 . Thus (2m + W - V)" 1 « l/2m - (w - V)/(2m) 2 (12-1) Then Eq. (11-7) becomes (W- V)* a = (l/2m)(cr-ir) 2 * a - (l/4m 2 )(or- ir)(W- V)(<r-ir)¥ a (12-2) while the normalizing requirement J(* a 2 + * b 2 ) d vol = 1, becomes /* a *[l + (a-ir) 2 /(4m 2 )]* a d vol = 1 (12-3) By use of the substitution X = [1 + (cr-7T)/(8m 2 )]* a (12-4) the normalizing integral can be simplified to read (to order v 2 /c 2 ) 52 QUANTUM ELECTRODYNAMICS J x *X d vol = 1 This substitution also allows easier interpretation of Eq. (12-2). Rewriting Eq. (12-2), [1 + ((7-7r) 2 /(8m 2 )](W-V)[l + (cr-7r) 2 /(8m 2 )]* a = (l/2m)(cr-7r) 2 * a + (l/8m 2 )[((7-7r) 2 (W- V) - 2(<r -ir)(W- V) x (0--7T) + (W- V)(a-ir) 2 ]* a Then applying Eq. (12-4) and dividing by 1 + (o"-ir) 2 /(8m 2 ), there results (W-V)x =(l/2m)((7-7r) 2 x- (l/8m 3 )(or-7r) 4 x + (l/8m 2 )[((7- 7T) 2 (W- V)-2((7'7T)(W-V)((7-7r) + (W-V)((7-7r) 2 ] X (12-5) The techniques of operator algebra may be used to convert Eq. (12-5) to a form more easily interpreted. In particular one should recall that A 2 B - 2 ABA + BA 2 = A(AB - BA) - (AB - BA)A Then, since ff = (p - eA), and since (C7-7T)(W- V) - (W-V)(cr-7r) = -i(cr • VV) = +i((7 -E) there results [with a-ir = A and (W- V) = B in the foregoing], i (a • it) (a • E) - i {a • E) (a • ir) - V • E + 2a • (it x E) (since V x E ~ 8B/9t = here), so Eq. (12-5) can be expanded as W X = Vx + (l/2m)(p - eA)-(p - eA) X - (e/2m)(<7 • B) x (1) (2) (3) -(l/8m 3 )(p-p) 2 X (4) + (e 2 /8m 2 )[V-E + 2a- (p - eA) x E] x (12-6) (5) C<5) In this form the wave equation may be interpreted by considering each term of Eq. (12-6) separately. Term (1) gives the ordinary scalar potential energy as it has appeared before. RELATIVISTIC WAVE EQUATION 53 Term (2) can be interpreted as the kinetic energy. Term (3), the Pauli spin effect, is just as it appears in the Pauli equa- tion. Term (4) is a relativistic correction to the kinetic energy. The correc- tion derives from E = (m 2 + p 2 ) 1/2 = m(l + p 2 /m 2 ) 1/2 = m + p 2 /2m - p 4 /8m 3 + '" The last term in this expansion is equivalent to term (4). Terms (5) and (6) express the spin-orbit coupling. To understand this in- terpretation consider the part of term (6) given by <j • (p x E). In an inverse- square field this is proportional to a • (p x r)/r 3 . The factor p x r can be interpreted as the angular momentum L to get (<j- L)/r 3 , the spin-orbit cou- pling. This term has no effect when the electron is in a s-state (L = 0). On the other hand, (5) reduces to V-E = 47rZ6(r), which affects only the s-states (when the wave function is nonzero at r = 0). So (5) and (6) together result in a continuous function for spin-orbit coupling. The magnetic moment of the electron e/2m, appears as the coefficient of term (3), and again of terms (5) and (6), i.e., (e/2m)(l/4m 2 ). A classical argument can be made to interpret term (6). A charge mov- ing through an electric field with velocity v feels an effective magnetic field B=vxE = (l/m)(p - eA) x E, and term (6) is just the energy (e/2m) x (<7-B) in this field. We get a factor 2 too much this way, however. Even be- fore the development of the Dirac equation, Thomas showed that this simple classical argument is incomplete and gave the correct term (6). The situa- tion is different for the anomalous moments introduced by Pauli to describe neutrons and protons (see Problem 3 below). In Pauli 's modified equation, the anomalous moment does appear with the factor 2 when multiplying terms (5) and (6). Problems: (1) Apply Eq. (12-6) to the hydrogen atom and correct the energy levels to first order. The results should be compared to the exact results. f Note the difference of the wave functions at the origin of coordinates. This difference actually is too restricted in space to have any importance. Near the origin the correct solution to the Dirac equation is proportional to r[l-(Z/13 7) 2 ] 1/2 «r- 1/40 ' 000 for the hydrogenic atoms, while the Schrodinger equation gives ^ — constant as r —*■ 0. tSchiff, "Quantum Mechanics," McGraw-Hill, New York, 1949, pp. 323ff. 54 QUANTUM ELECTRODYNAMICS (2) Suppose A and depend on time. Let W = i3/8t and follow through the procedures of this lecture to the same order of approxi- mation. (3) Pauli's modified equation can be applied to neutrons and pro- tons. It is obtained by adding a term for anomalous moments to the Dirac equation, thus y M (iV M - eA M )* +M7 M y y F Ml/ ^ = m* Multiplying by (3, this may be written in the more familiar "Hamil- tonian" expression i(8/8t)* = H Dirac * +ju0((7-B-a ■ E)* Show that the same approximation which led to Eq. (12-6) will now produce the terms [V + l/2M(p - eA) 2 + (jn +e/2M)(7-B + (l/8M 3 )(p- p) 2 + (1/4M 2 )(2 M +e/2M)(V-E + 2a- (p- eA)X E)] * (12-7) for protons, and a similar expression for neutrons, but with e = 0. (4) Equation (12-7) can be used to interpret electron-neutron scat- tering in an atom. Most of the scattering of neutrons by atoms is the isotropic scattering from the nucleus. However, the electrons of the atom also scatter, and give rise to a wave which interferes with nu- clear scattering. For slow neutrons, this effect is experimentally ob- served. It is interpreted by term (5) of Eq. (12-6) [as modified in Eq. (12-7) with e = 0] . Since the electron charge is present outside the nucleus, V- E has a value different from 0. Term (5) can be used in a Born approximation to compute the amplitude for neutron-electron scattering. However, when the effect was first discovered, it was ex- plained by the assumption of a neutron-electron interaction given by the potential c<5(R), where 6 is the Dirac 6 function and R is the neu- tron-electron distance. Compute the scattering amplitude with c6(R) by the Born approxi- mation and compare with that given by term (5). Show that c =47rjx N e 2 /4M N 2 In order to interpret c6(R) as a potential, the average potential V is defined as that potential which, acting over a sphere of radius e 2 /mc 2 , would produce the same effect. Using ji n = - 1.9135 eh/2M N , show that the resulting V agrees with experimental results within the stated accuracy, i.e., 4400 ±400 ev.f tL. Foldy, Phys. Rev., 87, 693 (1952) RELATIVISTIC WAVE EQUATION 55 (5) Neglecting terms of order v 2 /c 2 , show that /* f *af(R)*f d vol "* /Xf*[(pf +fp)/2m + ((7/2m)X(Vf)] Xi d vol Solution of the Dirac Equation for a Free Particle Thirteenth Lecture It will be convenient to use the form of the Dirac equation with the y 's when solving for the free-particle wave functions y M (iV^ - eA^)* = m* Using the definition of Lecture 10, # ^yu&u* * = y u A M = y t A t - y x A x - y y A y - y z A z V =r M V M =y t V t -y x V x -y y V y - y z V z and the Dirac equation may be written (iJP-e#)* =m* (13-1) (Recall that the quantity # = y^a^ is invariant under a Lorentz transforma- tion.) It is necessary to put the probability density and current into a four- dimensional form. In the special representation, the probability density and current are given by p = xjr*^ j = xj,* a ^ 56 SOLUTION OF THE DIRAC EQUATION 57 If the relativistic adjointj of * is defined *=**£ (13-2) in the standard representation, then the probability density and current may be written p = 5/3* j^ = 3?y M * To verify this, replace * by **/3 and note that /3 2 = 1 and that (Sy^ = a^. Exercises: (1) Show that the adjoint of * satisfies #(-iJP - e#) = m* (13-3) (2) From Eqs. (13-1) and (13-3) show that V^ j = (conservation of probability density). In general, the adjoint of an operator N is denoted by N, and N is the same as N except that the order of all y 's appearing in it is reversed, and each explicit i (not those contained in the y's) is replaced by -i. For ex- ample, if N = y x y y , N= y y y x = -N. If N = iy 5 = iy x y y y z y t , then N = -iy t y z y y y x = -iy 5 . The following property takes the place of the Hermitian property so useful in nonrelativistic quantum mechanics: (? 2 N*i)* = (*iN* 2 ) (13-4) For a free particle, there are no potentials, so $ = and the Dirac equation becomes i fi* = m* To solve this, try as a solution * =ue -ip-x = ue-iPyjc,, (13_ 6) t* is a four -component column vector, The adjoint * is the four-component row vector *j*, *2* ~^3* ~^4* in the standard representation. Multiplication by /3 changes the sign of the third and fourth components, in addition to changing ** from a column vector to a row vector. 58 QUANTUM ELECTRODYNAMICS * is a four -component wave function and what is meant by this trial solution is that each of the four components is of this form, that is, e -i P .x Thus Uj, u 2 , u 3 , and u 4 are the components of a column vector, and u is called a Dirac spinor. The problem is now to determine what restrictions must be placed on the u's and p's in order that the trial solution satisfy the Dirac equation. The V^ operation on each component of * multiplies each component by -ip„ , so that the result of this operation on ^ produces v u* =v u ue" i ^ x i; =-ip M ue- i Py x y - -ip M # so that Eq. (13-5) becomes iy M (-ip M )* = y M p M tf = #* = m* (13-7) Thus the assumed solution will be satisfactory if $u = rau, To simplify writing, it will now be assumed that the particle moves in the xy plane, so that Pi = Px P2 = Py P3 = ° P4 = E Under these conditions, $ = y t E - T y P y "TxPx- m standard representation / o\, „\ 1 1 -1 -1 7t I 0-1 J 7 *>v \-<7 x>y so i> -m becomes (13-8) By components, Eq. (13-7) becomes (E - mju! - (p x - ip y )u 4 = (13-9a) (E - m)u 2 - (p x + ip y )u 3 = (13-9b) (Px " iPy) u 2 - (E + m)u 3 = (13-9c) (p x + ip y )u 1 - (E + m)u 4 = (13-9d) SOLUTION OF THE DIRAC EQUATION 59 The ratio u t /u 4 can be determined from Eq. (13-9a) and also from Eq. (13-9d). These two values must agree in order that Eq. (13-6) be a solution. Thus ui/u 4 = (p x -ip y )/(E - m) = (E ■+ m)/(p x + ip y ) or 2 2 2 ^2 (13-10) p x 2 + p y 2 + m 2 = E 2 This is not a surprising condition. It states that the p v must be chosen so as to satisfy the relativistic equation for total energy. Similarly, Eqs. (13-9b) and (13 -9c) can be solved for u 2 /u 3 giving u 2 /u 3 = (p x + iPy)/(E - m) = (E + m)/(p x -ip y ) which also leads to condition (13-10). A more elegant way of obtaining exactly the same condition is to start directly with Eq. (13-7). Then, by multiplying this equation by $ gives $($u) = $(mu) = m($u) = m^ Using Eq. (10-9), ]^=p-p = E 2 -p x 2 -p y 2 so that the condition becomes E 2 - p x 2 - p y 2 = m 2 or u = The former is the same condition as obtained before, and the latter is a trivial solution (no wave function). Evidently there are two linearly independent solutions of the free-particle Dirac equation. This is so because substitution of the assumed solution, Eq. (13-6), into the Dirac equation gives only a condition on pairs of the u's, Uj, u 4 and u 2 , u 3 . It is convenient to choose the independent solutions so that each has two components which are zero. Thus the u's for the two solutions can be taken as and (13-11) where the following notation has been used: 60 > QUANTUM ELECTRODYNAMICS (13-12) F = E + m P + = Px + iPy P. = Px -ip y These solutions are not normalized DEFINITION OF THE SPIN OF A MOVING ELECTRON What do the two linearly independent solutions mean? There must be some physical quantity that can still be specified, which will uniquely deter- mine the wave function. It is known, for example, that in the coordinate sys- tem in which the particle is stationary there are two possible spin orienta- tions. Mathematically speaking, the existence of two solutions to the eigen- value equation ]6u = mu implies the existence of an operator that commutes with $ . This operator will have to be discovered. Observe that y 5 anticom- mutes with $; that is, y 5 $ = -jfy 5 . Also observe that any operator )N will anticommute with $ if W • p = 0, because W = -tifl + 2W-p (10-9) The combination y 5 J(Vof these two anticommuting operators is an operator which commutes with $; that is, (y 5 W = -y 5 ]6W=^(y 5 W) The eigenvalues of the operator (iygJ^O must now be found (the i has been added to make eigenvalues come out real in what follows). Denoting these eigenvalues by s, (iy 5 ^V)u = su (13-13) To find the possible values of s, multiply Eq. (13-13) by iy^, (iy 5 W)(iy 5 W)u = -y 5 Wy 5 Wu = -W-Wu = iy 5 W su = s\i or -W-W = s 2 If W»W is taken to be -1, the eigenvalues of the operator iy 5 )V are ± 1. The significance of the choice W • W = -1 is as follows: In the system in which the particle is at rest, p x = p y = p 2 =0 and p 4 = E. Then = p • W = p 4 W 4 or W 4 = SOLUTION OF THE DIRAC EQUATION 61 Thus, W • W = -W • W = -1 or W • W = 1. This states that in the coordinate system in which the particle is at rest, W is an ordinary vector (it has zero fourth component) with unit length. When the particle moves in the xy plane, choose )N to be y z , so the operator equation for iy 5 )V becomes iy 5 y z u =su Using relationships derived in Lecture 10, this becomes, for a stationary particle, f iy5Y*u = iy x y y y t u = iy x y y ii = - - \0 This choice makes fJ the a z operator, and the relationship with spin is clearly demonstrated. If we define u to satisfy both $u = mu and iy5^u = su, this completely specifies u. It represents a particle moving with momentum p^ and having its spin (in the coordinate system moving with the particle) along the W^ axis either positive (s = +1) or negative (s = -1). Exercise: Show that the first of the wave functions, Eq. (13-11), is the s = +1 solution and the second is the s = -1 solution. Another way of obtaining the wave function for a freely moving electron is to perform an equivalence transformation of the wave function as in Eq. (10-12). If the electron is initially at rest with its spin up or down in the z direction, then the spinor for an electron moving with a velocity v in the spatial direction k is u(k) =Su u - (2m) 1/2 u u = [ v j or [For normalization, see Eq. (13-14).] From Eq. (10-11), S is given by S = exp[(u/2)y t y k ] cosh u = 1/(1 - v 2 ) 1/2 Now exp[-(u/2)y t y k ] = cosh (u/2) + y t y k sinh (u/2) t For a stationary particle y t u = u. 62 ^ QUANTUM ELECTRODYNAMICS and (2m) 1/2 cosh (u/2) = [m(l - v 2 )" 1/2 + m] 1 / 2 = (E + m) (2m) 1/2 sinh (u/2) = (E - m) 1/2 Therefore, u (k) = [<E + m) 1/2 + y t y k (E - m) 1/2 ] u Writing f = (E + m), a = y t y, and noting (E 2 -m 2 ) 1/2 =p k> we get U( k) = (l/VF)(E+m+a -p)u For the case that p is in the xy plane, this just gives the result, Eq. (13-11) with a normalization factor 1/Vf. Noticing that for an electron at rest y t u = u , u (k) may be written (l/VF)(Ey t -y.'P + m)u or u (k) = (l/VF)(rf + m)u It is clear that this is a solution to the free-particle Dirac equation (# - m)u k = (13-7) for $ + m)(^ - m) = p 2 - m 2 = p 2 = m 2 NORMALIZATION OF THE WAVE FUNCTIONS In nonrelativistic quantum mechanics, a plane wave is normalized to give unity probability of finding the particle in a cubic centimeter, that is, \j?*\ff = l. An analogous normalization for the relativistic plane wave might be some- thing like \£*>j, = u * u = uy t u = 1 However, ^*^ transforms similarly to the fourth component of a four- vector (it is the fourth component of four-vector current), so this normal- ization would not be invariant. It is possible to make a relativistically in- variant normalization by setting u* u equal to the fourth component of a SOLUTION OF THE DIRAC EQUATION 63 suitable four-vector. For example, E is the fourth component of the mo- mentum four-vector p^, so the wave function could be normalized by uy t u = 2E The constant of proportionality (2) is chosen for convenience in later for- mulas. Working out (uy t u) for the s = + 1 state, 1 °\ r 1 1 p -1 -1/ \p x C^ x Cj 2 = (F 2 + p + pJCi 2 = 2E(E + m)Ci 2 The Cj is the normalizing factor multiplying the wave functions of Eq. (13-11). In order that (uy t u) be equal to 2E, the normalizing factor must be chosen (E + m)" 1/2 = (F)" 1 ' 2 . In terms of (uu), this normalizing condi- tion becomes (uu) = F -p — /T?2 (F'-p_p + ) 2m 2 + 2mE E + m 2m The same result is obtained for the s = -1 state. Thus the normalizing condition can be taken as (uu) = 2m (13-14) In a similar manner, the following can be shown to be true: (uy x u) = 2p x (uy y u) = 2p y (uy z u) = It will be convenient to have the matrix elements of all the y's between va- rious initial and final states, so Table 13-1 has been worked out. 64 QUANTUM ELECTRODYNAMICS TABLE 13-1. Matrix Elements for Particle Moving in the xy Plane Matrix N s = 1 2m 7x 2p x ?y 2p y Tz Vt 2E VF^gCuaNu!) VF 1 F 2 (u 2 Nu 1 ) Vl^ ( u 2 Nu t ) VF 1 F 2 (u 2 Nu 1 ) (uNu) S! = +l Sj= + 1 St = -1 S! = -1 s 2 = + 1 s 2 = -1 s 2 = -1 S, = + 1 F 2 F 1 - Pi + P2- F2P1+ + P2-F! -iF 2 Pi++ ip 2 -F! -P1 + F2+P2+FJ F 2 F 1 + Pi + p 2 _ TyTz -iF 2 F,+ ip 1 + p 2+ TzYx F 2 F 1 + p 1+ p 2+ y x y y -2iE -iF 2 F 1 -ip 1 + p 2 _ y t y x 2i Py F 2Pl+ -p 2 _F 1 y t y y -2i Px -iFup^-ip^Fi yty z -pi+Fz-pz+f, y5y x =yty y yz y5y y =y t yzy x o ysy z = y t y x y y -2im ysy t =y x y y yz o -iF 2 F t -ipi + p 2 + FgFj - pi + p 2 + ; iF 2 F 1 +-ip 1+ p 2 _ iF 2 Pi + +iF lP2+ «H CO + II «T a ei <4-H fafi O 3 CD O O 3 X C? O O a O s X a> a <4H S O CD O > d bD CD fe y5 = y x y y y z y t ° iF 2 Pi+ - iF!p 2 + Note: p 2+ = p 2x + iP2y = P2 exp (i0 2 ); P 2 - = P2 X ~ iP2 y = P2 exp (-i0 2 ); F 2 = E 2 + m; F, = Ej + m; p 2 = (E - m)F. SOLUTION OF THE DIRAC EQUATION 65 Limiting cases: To obtain the case where 1 is a positron at rest, the table gives VF2(u"2 Nu i) if one P uts F A = 0, p 1+ = 1 = Pj_ in the table. For both at rest as positrons, the table gives (u^Nuj) with F { = F 2 = 0; p 1+ =p 2+ = 1. Fourteenth Lecture METHODS OF OBTAINING MATRIX ELEMENTS The matrix element of an operator M between initial state u 1 and final state u 2 will be denoted by (u^Muj) The matrix element is independent of the representations used if they are related by unitary equivalence transformations. That is, u'j = Su t u' 2 = Su 2 M' = SMS" 1 u" 2 = a 2 s so that a^M'u'i - u 2 SS MS^Sut = UgMu! where the property S = S _1 has been assumed for S. The straightforward method to compute the matrix elements is simply to write them out in matrix form and carry out the operations. In this way the data in Table 13-1 were obtained. Other methods may be used, however, sometimes simpler and sometimes leading to corollary information, as illustrated by the following example. By the normalization convention, flu = 2m Hence (fl^u) = 2m 2 since $u = mu. Similarly, 66 QUANTUM ELECTRODYNAMICS (fly^tfu) =m(tfy M u) But also note that (uf*y M u) = m(fly M u) because fl$ = $fl = mu. Adding the two expressions, one obtains (fl(y M tf+l%)u) =2m(u> M u) From the relation proved in the exercises that = -W + 2a * b it is seen that *% + y u t =2p u y p =x But p is just a number, so it follows that 2p p (flu) = 2m(fly p u) and since flu = 2m, by normalization (tfy M u)=2p M Furthermore, the general relation (fly t u)/(flu) = p 4 /m = E/m is obtained. From this it is seen why the possible normalization (fly^u) * E/m was equivalent to (flu) = 1. Problem: Using methods analogous to the one just demonstrated, show that (fly 5 u) =0 INTERPRETATIONS OF NEGATIVE ENERGY STATES It was found that a necessary condition for solution of the Dirac equation to exist is E 2 = p 2 + m 2 SOLUTION OF THE DIRAC EQUATION 67 E =±(p 2 + m 2 ) 1/2 The meaning of the positive energy is clear but that of the negative is not. It was at one time suggested by Schrodinger that it should be arbitrarily ex- cluded as having no meaning. But it was found that there are two fundamen- tal objections to the exclusion of negative energy states. The first is physi- cal, theoretically physical, that is. For the Dirac equation yields the result that starting with a system in a positive energy state there is a probability of induced transitions into negative energy states. Hence if they were ex- cluded this would be a contradiction. The second objection is mathematical. That is, excluding the negative energy states leads to an incomplete set of wave functions. It is not possible to represent an arbitrary function as an expansion in functions of an incomplete set. This situation led Schrodinger into insurmountable difficulties. Problem: Suppose that for t <0 a particle is in a positive en- ergy state moving in the x direction with spinup in the z direction (s = + 1). Then at t = 0, a constant potential A = A z (A x = A y = 0) is turned on and at t = T it is turned off. Find the probability that the particle is in a negative energy state at t = T. Answer: Probability of being in | negative energy state > = A 2 /(A 2 + m 2 ) sin 2 [(m 2 + A 2 ) 1/2 T] at t = T J Note that when E = -m, 1/VF" = °° , so the u's apparently blow up. But actually the components of u also vanish when E = — m, so that a limiting process is involved. It may be avoided and the correct results obtained simply by omitting 1/VF and replacing F by zero and p ± by 1 in the components of u. The positive energy levels form a continuum extending from E = m to +°°, and the negative energies if accepted as such form another continuum from E = — m to -°° . Between +m and — m there are no available energy levels (see Fig. 14-1). Dirac proposed the idea that all the negative energy levels are normally filled. Explanations for the apparent obscurity of such a sea of electrons in negative energy states, if it exists, usually contain a psycho- logical aspect and are not very satisfactory. But, nevertheless, if such a situation is assumed to exist, some of the important consequences are these: 1. Electrons in positive energy states will not normally be observed to make transitions into negative energy states because these states are not available; they are already full. 2. With the sea of electrons in negative energy levels unobservable, a "hole" in it produced by a transition of one of its electrons into a positive energy state should manifest itself. The manifestation of the hole is re- garded as a positron and behaves like an electron with a positive charge. 68 QUANTUM ELECTRODYNAMICS + 00 A + m positive energy levels -m negative energy levels normally filled FIG. 14-1 3. The Pauli exclusion principle is implied in order that the negative sea may be full. That is, if any number rather than just one electron could oc- cupy a given state, it would be impossible to fill all the negative energy states. It is in this way that the Dirac theory is sometimes considered as ''proof" of the exclusion principle. Another interpretation of negative energy states has been proposed by the present author. The fundamental idea is that the "negative energy" states represent the states of electrons moving backward in time. In the classical equation of motion m(d 2 z M /ds 2 ) =e(dz y /ds)F My reversing the direction of proper time s amounts to the same as reversing the sign of the charge so that the electron moving backward in time would look like a positron moving forward in time. In elementary quantum mechanics, the total amplitude for an electron to go from x^tj to x 2 ,t 2 was computed by summing the amplitudes over all possible trajectories between K\,t\ and x 2 ,t 2 , assuming that the trajec- tories always moved forward in time. These trajectories might appear in one dimension as shown in Fig. 14-2. But with the new point of view, a pos- sible trajectory might be as shown in Fig. 14-3. Imagining oneself an observer moving along in time in the ordinary way, being conscious only of the present and past, the sequence of events would appear as follows: SOLUTION OF THE DIRAC EQUATION 69 FIG. 14-2 FIG. 14-3 1— *P p-t- — *t< only the initial electron present the initial electron still present but somewhere else an electron-positron pair is formed the initial electron and newly arrived electron and positron are present the positron meets with the initial electron, both of them annihilating, leaving only the previously created electron only one electron present To handle this idea quantum mechanically two rules must be followed: 70 QUANTUM ELECTRODYNAMICS 1. In calculating matrix elements for positrons, the positions of the ini- tial and final wave functions must be reversed. That is, for an electron mov- ing forward in time from a past state * past to a future state ^ fut , the ma- trix element is / $ ( . Mtf „ d vol J fut oast fut past But moving backward in time, the electron proceeds from ¥ firf to * nast so the matrix element for a positron is i\ast M *fut dvo1 2. If the energy E is positive, then e~ ip ' x is the wave function of an elec- tron with energy p 4 = E. If E is negative, e~ ip ' x is the wave function of a positron with energy -E or |E|, and of four -momentum -p. Potential Problems in Quantum Electrodynamics Fifteenth Lecture PAIR CREATION AND ANNIHILATION Two possible paths of an electron being scattered between the states ^ t and ^2 were discussed in the last lecture. These are: Case I. Both * lf * 2 states of positive energy, interpreted as ^i electron in ''past," #2 electron in "future." This is electron scattering. Case II. Both ¥ lf * 2 states of negative energy interpreted as ^ posi- tron in "future," \£ 2 positron in "past." This is positron scattering. The existence of negative energy states makes two more types of paths possible. These are: Case III. The ^ positive energy, * 2 negative energy, interpreted as 4^ in "past," ^2 positron in "past." Both states are in the past, and nothing in the future. This represents pair annihilation. Case IV. The #j negative energy, * 2 positive energy, interpreted as \frj positron in "future," * 2 electron in "future." This is pair creation. Case IV 71 72 QUANTUM ELECTRODYNAMICS The four cases can be diagrammed as shown in Fig. 15-1. Note that in each diagram the arrows point from % to ^ although time is increasing upward in all cases. The arrows give the direction of motion of the elec- tron in the present interpretation of negative energy states. In common lan- guage, the arrows point toward positive or negative time according to whether $ is positive or negative, that is, whether the state represented is that of an electron or a positron. CONSERVATION OF ENERGY Energy relations for the scattering in case I have been established in previous lectures. It can be seen that identical results hold for case II. To show this, recall that in case I, if the electron goes from the energy E t to E 2 and if the perturbation potential is taken proportional to exp(— icot), then this perturbation brings in a positive energy go. To see this, note that the amplitude for scattering is proportional to /exp(-iE 2 t)* exp(-io;t) exp(-iE 1 t) dt = /exp[(iE 2 t - iwt - iE t t) dt] (15-1) As has been shown, there is a resonance between E 2 and Ej -fa;, so that the only contributing energies are those for which E 2 « E± + oj. In case II the same integral holds but E 2 and E t are negative. A positron goes from an energy (past) of E past = -E 2 to an energy (future) of E fut = -E t . With the same perturbation energy, the amplitude is large again only if E 2 = Ej + co or -E past = -E fut + u), so that E fut = co + E past ; that is, the perturbation car ries in a positive energy co, just as it does for the electron case. THE PROPAGATION KERNEL In the nonrelativistic case (Schrodinger equation), the wave equation, in- cluding a perturbation potential, is written i9*/at = H *+ V* (15-2) where V is the perturbation potential and H is the unperturbed Hamilto- nian. For the free particle, the kernel giving the amplitude to go from point 1 to point 2 in space and time can be shown to be K (2,l) =Nexp[(l/2)im(x 2 -x 1 ) 2 /(t 2 -t 1 )] t 2 >t t = t 2 < t A (15-3) where N is a normalizing factor depending on the time interval t 2 - tj and the mass of the particle: PROBLEMS IN QUANTUM ELECTRODYNAMICS 73 N = [m/2iri(t 2 - t t )] 1/2 Note that the kernel is defined to be for t 2 < t t . It can be shown that K satisfies the equation [18/8 1 2 - H (2)] K (2,l) = 16(2,1) (15-4) The propagation kernel K v (2,l) giving a similar amplitude, but in the presence of the perturbation potential V, must satisfy the equation [18/8 1 2 - H (2) - V(2)] K v (2,l) = 16(2,1) (15-5) It can be shown that Ky can be computed from the series K V (2,D = K (2,l) - i /Ko(2,3)V(3)Ko(3,l)d 3 x 3 dt 3 - /K (2,4)V(4)K (4,3)V(3)K (3,l)d 3 x 4 dt 4 d 3 x 3 dt 3 + •■ (15-6) In case the complete Hamiltonian H = H + V is independent of time, and all the stationary states n of the system are known, then K v (2,l) may be obtained from the sum K v (2,l) =2 exp[-iE n (t 2 - t^] n (x 2 )0 n *( Xl ) (15-7) n The extension of these ideas to the relativistic case (Dirac equation) is straightforward. By choosing a particular form for the Hamiltonian, the Dirac equation can be written i8*/8t = H* = a ■ (p - eA)* + e<f) * + m/S* Defining the propagation kernel as K A , then the kernel is the solution to the equation [i8/8t 2 - e0 2 - a • (-1V- eA 2 ) - mfi] K A (2,1) = i/36(2,l) (15-8) The matrix (3 is inserted in the last term in order that the kernel derived from the Hamiltonian be relativistically invariant. [Note the similarity to the nonrelativistic case, Eq. (15-6).] Multiplying this equation by /3, a sim- pler form results: {if 2 " e£ 2 - m)K A (2,l) = 16(2,1) (15-9) The equation for a free particle is obtained simply by letting A 2 = 0, then calling the free-particle kernel K + , 74 QUANTUM ELECTRODYNAMICS (ift- m)K + (2,l) =16(2,1) (15-10) The notation K + replaces the K of the nonrelativistic case, and Eq. (15-10) replaces Eq. (15-4) as the defining equation. Just as K v can be expanded in the series of Eq. (15-6), so K A can be expanded as K A (2,1) = K + (2 f l) -i/K + (2,3)e#(3)K + (3,l)dT 3 -/K + (2,3)e^(3)K + (3,4)e^(4)K + (4,l)dT 3 dr 4 + ••• (15-11) Note that the kernel is now a four-by-four matrix, so that all components of ^ can be determined. Since this is true, the order of the terms in Eq. (15-11) is important. The element of integration is actually an element of volume in four-space, dT = dxi dx 2 dx 3 dx 4 The potential, -ie$(l) can be interpreted as the amplitude per cubic centi- meter per second for the particle to be scattered once at the point (1). Thus the interpretation of Eq. (15-11) is completely analogous to that of Eq. (15-6). Problem: Show that K A as defined by Eq. (15-11) is consistent with Eqs. (15-8) and (15-9). On the nonrelativistic case, the paths along which the particle reversed its motion in time are excluded. In the present case this is no longer true. The existence and interpretation of the negative energy eigenvalues of the Dirac equation allows the interpretation and inclusion of such paths. Taking t 4 > t 3 implies the existence of virtual pairs. The section from t 4 to t 3 represents the motion of a positron (see Fig. 15-2). In a time -stationary field, if the wave functions <p n are known for all the states of the system, then K + A may be defined by K + A (2,l) = 23 exp[-iE n (t 2 -t 1 )]0 n (x 2 )0 n (x 1 ) pos. energies t 2 >t, 2 exp[-iE n (t 2 -t 1 )]0 n (x 2 )0 n (x 1 ) neg. energies t 2 < tj (15-12) PROBLEMS IN QUANTUM ELECTRODYNAMICS 75 FIG. 15-2 Another solution of Eq. (15-9) is K M2,1) = £ exp[-iE n (t 2 -t 1 )]0 n (x 2 )? n (x 1 ) pos. energies +■ 2 exp[-iE n (t 2 -t 1 )]0 n (x 2 )0 n (x 1 ) t 2 >tj neg. energies to < ti (15-13) Equation (15-13) has an interpretation consistent with the positron inter- pretation of negative energy states. Thus when the timing is "ordinary" (t 2 > tj), an electron is present, and only positive energy states contribute. When the timing is "reversed" (t 2 < tj), a positron is present, and only negative energy states contribute. On the other hand, Eq. (15-13) does not have so satisfactory an interpretation. Although the kernel K A defined by Eq. (15-13) is also a satisfactory mathematical solution of Eq. (15-9) (as shown below), the interpretation of Eq. (15-13) requires the idea of an elec- tron in a negative energy state. To show that both kernels are solutions of the same inhomogeneous equa- tion, note that their difference is £ exp(iE n tt) exp(-iE n t 2 )0 n (x 2 )0 n (x 1 ) neg. energies for all t 2 . This is, term by term, a solution of the homogeneous equation [i.e., Eq. (15-9) with zero right-hand side]. The possibility that two such 76 QUANTUM ELECTRODYNAMICS solutions exist results from the fact that boundary conditions have not been definitely fixed. We shall always use K + A . The kernel K + A , defined by Eq. (15-12), allows treatment of case III (pair annihilation) and case IV (pair creation) shown at the beginning of this lec- ture. In each case, the potential, -ie$(3), acts at the intersection of positron and electron paths. Sixteenth Lecture USE OF THE KERNEL K + (2,l) In the nonrelativistic theory it was possible to calculate the wave function at a point x 2 at time t 2 from a knowledge of the wave function at an earlier time t t (see Fig. 16-1) by means of the nonrelativistic kernel K (x2,t 2 ; x^tj), *(x 2 ,t 2 ) = /K (x 2 ,t 2 ; xi,t 1 )*(x 1 ,t 1 ) d 3 X! It might be expected that a relativistic generalization of this would be *(x 2 ,t 2 ) = /K + (x 2 ,t 2 ; x 1 ,t 1 )y t ^(x 1 ,t 1 ) d 3 Xl U ft • x 2 ,t 2 tl 1 *» • Xo.t 2> u 2 -►x FIG. 16-1 FIG. 16-2 This turns out to be incorrect, however. It is not sufficient, in the relativis- tic case, to know just the wave function at an earlier time only because K + (2,l) is not zero for t 2 < tj. When the kernel is defined in this manner (Lecture 15), the wave function at x 2 ,t 2 (see Fig. 16-2) is given by *(x 2 ,t 2 ) = /K + (x 2 ,t 2 ; x 1 ,t 1 )y t ^(x 1 ,t 1 )d 3 x 1 - /K + (x 2 ,t 2 ; x ll t 1 ')-yt*<x 1 .ti') Al tt< t 2 < V (16-1) PROBLEMS IN QUANTUM ELECTRODYNAMICS 77 The first term is the contribution from positive energy states at earlier times and the second term is the contribution from negative energy states at later times. This expression can be generalized to state that it is nec- essary to know ^(Xitj) on a four-dimensional surface, surrounding the point x 2 ,t 2 (see Fig. 16-3): *(x 2 ,t 2 ) = /K + (2 f 1)#(1)¥(1) d 4 Xl (16-2) where $ is the four-vector normal to the surface that encloses x 2 ,t 2 . FIG. 16-3 TRANSITION PROBABILITY The amplitude to go from a state f to a state g under the action of a po- tential jL is given by an expression similar to that in nonrelativistic theory, a 2 l = //g(2)j8K + A (2,l) i Sf(l)d 3 x 1 d 3 x 2 (16-3) Using the expansion of K + A (2,l) in terms of K + (2,l), Eq. (15-12), and assu- ming that the amplitude for transition from state f to state g as a free par- ticle is zero (f and g orthogonal states), the first-order amplitude for transition (Born approximation) is a 21 = -i /g(2) j3/K + (2,3)e^(3)K + (3,l)/3f(l)dT 8 d 3 Xl d 3 x 2 It is convenient to let f(3) = /K + (3,l)i3f(l)d 3 x 1 g(3) = /g(2)/3K + (2,3)d 3 x 2 These state that the particle has the free-particle wave function f just prior to scattering and the free-particle wave function g just after scattering, and that it eliminates any computation of the motion as a free particle. The am- plitude for transition, to first order, may be written 78 QUANTUM ELECTRODYNAMICS -i/g(3)e£(3)f(3)dT 3 (16-4) (dT 3 signifies integration over time as well as space). The second-order term would be written -(1/2) //g(4)M(4)K + (4 f 3)e£(3)f<3) dr 3 dr 4 If f(3) is a negative energy state, then it represents a positron of the future instead of an electron of the past and the process described by this ampli- tude is that of pair production. SCATTERING OF AN ELECTRON FROM A COULOMB POTENTIAL We shall make use of the theory just presented to calculate the scattering of an electron from an infinitely heavy nucleus of charge Ze. Suppose the incident electron has momentum in the x direction and the scattered elec- tron has momentum in the xy plane (see Fig. 16-4): A = 7t E i "TxPix jri 2 = y t E z -y x p 2x -y y p 2y FIG. 16-4 The potential is that of a stationary charge Ze, = Ze/r, A = # = y t (Ze/r) The initial and final wave functions are plane waves: f(l) = u 1 e~ ip r x g(2) =u 2 e~ ip 2* x (four -component wave function) Thus, by Eq. (16-4), the first-order amplitude for transition from state f to state g (momentum p t to momentum p 2 ) is M = -i/u 2 e lp 2-* (Ze 2 /r)y t u 'l e "' ,pi " x d3x dt PROBLEMS IN QUANTUM ELECTRODYNAMICS 79 Separating space and time dependence in the wave functions, this becomes M = -i(u 2 y t u 1 ) /e- i P2' x (Ze 2 /r)e i Pr x d 3 x / e^ e' iE ^ dt The first integral is just V(Q), a three-dimensional Fourier transform of the potential, which was evaluated in nonrelativistic scattering theory: I i(E 2 - E { ) J (16-5) V(Q) = 47rZe 2 /Q 2 Q = Pi - p 2 The probability of transition per second is given by Trans, prob./sec = 27r(nN) _1 |M| 2 x (density of final states) (16-6) This is a result from time -dependent perturbation theory, the only new fac- tor is a normalizing factor (IIN)" 1 which takes account for the fact that the wave functions are not normalized to unity per unit volume. The IIN is a product of factors N one for each wave function, or particle in the initial state, and one for each final wave function, N = (uy t u) (16-7) for each particle in question. In our normalization, then N - 2E. The reason for this factor is that wave functions are normalized to (uu) = 2m or (uy t u) = 2E where, as in the computation of transition probability, they should be nor- malized in the conventional nonrelativistic manner ty*^ = 1 or (uy t u) = 1 (so N = 1 for that case). The matrix element M, as calculated in this manner, is relativistically invariant and in the future the chief interest will be in M. The transition probability, knowing M, can be computed from Eq. (16-6). Density of States, Cross Section. For the electron scattering problem under consideration, M = -i(u 2 y t u 1 )(47rZe 2 /Q 2 ) so the transition probability is Trans. 27T prob./sec (2E t )(2E2) !(u 2 y t ui)| 47rZe' Q 2 E2P2dS2 (2tt) 3 (16-8) 80 QUANTUM ELECTRODYNAMICS where the density of final states has been obtained in the following manner: Density of states = 1T ^r = 4^3 but E 2 2 = p 2 2 + ni 2 , so dp 2 /dE 2 = E 2 /p 2 and (27r) 3 dE 2 (27r) 3 dE 2 R " 1 Density of states = E 2 p 2 dfi/(27r) 3 When the incoming plane wave is normalized to one particle per cubic cen- timeter, the cross section is given in terms of the transition probability per secondt as Trans. prob./ sec =crv 1 == o^pj/Ej) or a = (Ej/Pi) x (trans. prob./ sec ) The essential difference between the relativistic treatment of scattering and the nonrelativistic treatment is contained in the matrix element (u^Uj). From Table 13-1, for a particle moving in the xy plane and s 1 = + 1, s 2 = +1, l(u 2 7 t ui)| 2 = 1/F 1 F 2 |F 2 F 1 + p 1 + p 2 _| 2 where F 1 = F 2 = E + m [Ej = E 2 , conservation of energy, follows from the nature of the time integral in Eq. (16-5)] , and Pi+ =P P2- = Pe" i0 (magnitude of final momentum equal to magnitude of initial momentum fol- lows from E} = E 2 ). Thus mvj m 2 v/ t Pi = n v 2x1/2 ~* Pi 2 = i v 2 — Pi 2 = ( m2 + Pl 2 ) v i 2 = E l V (1 — Vj ) 1 - Vj Therefore, v A = Pi/Ej. PROBLEMS IN QUANTUM ELECTRODYNAMICS 81 l(u 2 7tUi)| 2 = (E + m)- 2 |(E+m) 2 + p 2 e - ie | 2 = (E ' + m) -2 {4E 2 (E +m) 2 [l - (p 2 /E 2 ) sin 2 (6/2)]} = (2E) 2 [1 - v 2 sin 2 (6/2)] When s 4 = +1, s 2 = -1 or Sj = -1, s 2 - + 1, the matrix element of y t is zero. When s 4 = -1, s 2 = -1, the absolute value of the matrix element is the same as for s t = +1, s 2 = +1. Thus spin does not change in scattering (in Born approximation) and the cross section is independent of spin, a = (4Z 2 e 4 E 2 /Q 4 )d<3 [1 - v 2 sin 2 (6/2)] Q = 2p sin (6/2) The criterion for validity of the Born approximation, used in obtaining this result, is Ze 2 /hv«l. In the extreme relativistic limit v~c. This becomes Z « 137. Just as for the nonrelativistic case, the scattering can actually be calculated exactly (correct to all orders in the potential) for the Coulomb potential. This exact solution of the Dirac equation involves hype rgeome trie functions. It was first worked out by Mott and is called Mott scattering. For moderate energies (200 kev) there is some probability for change in spin. Polarized electrons could be produced in this manner. Problems: (1) Calculate the Rutherford scattering law for the Klein-Gordon equation (particle with no spin). Result: same formula as just given with 1 - v 2 sin 2 (6/2) replaced by 1. (2) Show that this scattering formula is also correct for positrons (use positron states in calculating matrix element). Seventeenth Lecture CALCULATION OF THE PROPAGATION KERNEL FOR A FREE PARTICLE As shown in a previous lecture, the propagation kernel, when there is no perturbing potential and the Hamiltonian of the system is constant in time, is K + (2,l) =E^n(x 2 )0 n (Xi) exp[-iE n (t 2 - ti)] t 2 > tj + n = -S^n(x 2 )?„(x 1 )exp[-iE n (t 2 - ti)] t 2 < tj For a free particle, the eigenf unctions <p n are u p exp (ip-x) 82 QUANTUM ELECTRODYNAMICS and the sum over n becomes an integral over p. The u p is the spinor cor- responding to momentum p, positive or negative energy and spin up or down, as appropriate. Then the propagation kernel for a free particle is, for t 2 > tj^, d 3 p 1 (2tt) 3 2E_ spins v ' P K + (2,l) = £ / 7^f £jf UpUpexptip^Xg-Xi)] xexp[-iE p (t 2 - t t )] for E p = + (p 2 + m 2 ) 1/2 . The factor l/(27r) 3 is the density of states per unit volume of momentum space per cubic centimeter. The factor 1/2E arises from the normalization uu = 2m or u-y t u = 2E p used here. The Up. are those for positive energy. For negative energy E p = - (p 2 + m 2 ) 1/2 , the Up are changed accordingly and K + (2, 1) becomes, for t 2 < tj, K + (2,l) = - E/t^3 ^T UpU p exp[ip-(x 2 - Xl )] spins \ ' P xexp[iE p (t 2 - t t )] The calculation will be made first for the case of t 2 > t t . We first calcu- late Up Up for positive energy, and p in the xy plane and spin up. Under these conditions E +m / (E+m} l/2 Px + iPy 1 (E + m) 1 Note that u p u p is the opposite order to that usually encountered so that the product is a matrix, not a scalar. That is, (E+m) 2 (E+m)(-p x +ip y ) (E+m)(p x +ip y ) (Px+ip y )(-p x +iPy) x 1/(E + m) by the usual rules for matrix multiplication. But (Px + iPy)(-Px + iPy) = -P 2 = -E 2 + ™ 2 and the matrix becomes PROBLEMS IN QUANTUM ELECTRODYNAMICS 83 E + m Px + iPy By the same process, the result in the spin down case is -Px + iPy - E+ m (spin up) (E +m) 1 / 2 u p u p = (spin down) It may be verified easily that the sum of these matrices for spin up and spin down is represented by E Tt - PxTx - PyTy m In the general case when p is in any direction, it is clear that the only change is an additional term -p z y z . So, in general, (u p u) spin up + (u p u p ) spin down = Ey t -p-y + m = ^+m The sign of the energy was not used in obtaining this result so it is the same for either sign. Now put t2 - tj = t and x 2 - Xj = x. For t > 0, the propagation kernel be- comes K + (2, 1) = /(E p y t - p y + m)[d 3 p/(27r) 3 ](l/2E p ) x exp[-i(E p t - p-x)] The appearance of p in the form E p = (p 2 + m 2 ) 1/2 in the time part of the exponential makes this a difficult integral. Note that it may also be written in the form K + (2,i)=(iy tii+ iY x ^ + ir y - d 3 P _9_ 8z / (2^) 3 2E r expt-i(E p t-p.x)] = i(i? + m) I + (t,x) 84 QUANTUM ELECTRODYNAMICS where I + (t,x) =-i/ d^p_ (2tiT2E. exp[-i(E p t P-x)] In this form only one integral instead of four need be done. It may be veri- fied as an exercise that for t < the result is the same except that the sign of t is changed, so that putting |t| in place of t in the formula for I + (t,x) makes it good for all t. This integral has been carried out with the following result: I + (t,x) ="(4tt)- 1 6(s 2 ) + (m/87rs)H 1 (2) (ms) where s = + (t 2 delta function and Hj the foregoing is x 2 ) 1/2 for t > x, and -i(x 2 - t 2 ) 1/2 for t < x. 5(s ) is a (2) (ms) is a Hankel function.! Another expression for I + (t,x) = -(1/87T 2 ) J o °° da exp {-(i/2)[(m 2 /a0 + a(t 2 -x 2 )]} Both of these forms are too complicated to be of much practical use. It will be shown shortly that a tremendous simplification results from transforma- tion to momentum representation. Note that I + (t,x) actually depends only on |x|, not on its direction. In the time-space diagram (Fig. 17-1) the space axis represents |x| and the diag- onal lines represent the surface of a light cone including the t axis, that is, the accessible region of t - |x| space in the ordinary sense. It can be shown that the asymptotic form of I + (t,x) for large s is proportional to e~ ims . When one's region of accessibility is limited to the inside of the light cone, large s implies t 2 » |x| 2 , so that the region of the asymptotic approxima- tion lies roughly within the dotted cone around the t axis and is regions of asymptotic approximation surface of light cone (here I + is singular) Hxl FIG. 17-1 tSee Phys. Rev., 76, 749 (1949); included in this volume. PROBLEMS IN QUANTUM ELECTRODYNAMICS 85 I + (t,x) — e" ims w exp{-im[t - (x 2 /2t)]} * e" Imt The first form is seen to be essentially the same as the propagation kernel for a free particle used in nonrelativistic theory. If, as in the new theory, possible "trajectories" are not limited to regions within the light cone, an- other region included in this asymptotic approximation is that within the dotted cone along the |x| axis where large s implies |x| 2 »t 2 . Hence I + (t,x) — e~ ims = exp[-im(x 2 -t 2 ) 1/2 ] « e- m l x l It is seen that the distance along |x| in which this becomes small is roughly the Compton wavelength (recall that m — - mc/fi when it represents a length" 1 as here), so that in reality not much of the t — |x| space outside the light cone is accessible. The transformation to momentum representation will now be made. This is facilitated by use of the integral formula r oo exp (-ip 4 t) 7ri lim J dp 4 — o — ZTi ~ ' - - — exp (-iE_ t ) 6 ^ J -°° ^ 4 P 4 2 "E p 2 +i€ E p ^ v V\ " The ie term in the denominator is introduced solely to ensure passage around the proper side of the singularities at p 4 2 = E p 2 along the path of integration. Passage on the wrong side will reverse the sign in the exponential on the right. Problem: Work out the integral above by contour integration or otherwise. Using the integral relation above, I + (t,x) becomes t/ ^ r d 3 p exp(-ip 4 t) exp(+ip-x) I + (t.X) = J —< dp 4 pt 2_ 2 +u But E 2 = p 2 + m 2 so this is t / + x _ f d4 P exp[-i(p>x)] I+(t ' X) " J (27T) 4 p 2 - m 2 + ic 2 _ where p is now a four-vector so that d 4 p = dp 4 dp t dp 2 dp 3 , and p- = p^Py. Hereafter the ie term will be omitted. Its effect can be included simply by imagining that m has an infinitesimal negative imaginary part. In this form the transformation to momentum representation is easily accom- plished as follows (we actually take Fourier transform of both space and time, so this is really a momentum-energy representation): 86 QUANTUM ELECTRODYNAMICS i + (p) = /l + (t,x) exp[+i(p-x)]d 4 x r d 4 4 d 4 x exp[-i(£-p)- x] J (2tt) 4 £ 2 - m 2 where the dummy variable 4 has been substituted for p in the p integral. But J"exp[-i(£ -p) • x] d 4 x = (2tt) 4 6(£ -p) Hence the £ integration gives the result i + (p) = l/(p 2 -m 2 ) Finally, applying the operator i(i^ + m) to I + (t,x) gives the propagation ker- nel (here x = x 2 - x A ) K + (2,l) = W + m)I + (t,x) = i fj^i (JJT + m) ^'^^ . f d 4 p f) + m . .. v , = 1 J^ P^? exp[-i(p-x)] recalling that ip operating on exp[-i(p-x)] is the same as multiplying by $. From the identity 1 1 ^ + m _ j> + m i> - m $ - m $ + m p 2 -m 2 the kernel can also be written K , 9 n _ * T J^p_ exp[-i(p-x)] K +<2,D-i J (2*) 4 jj-m By the same process used for I + (t,x), the transform of K + (2,l) in momen- tum representation is seen to be k(p) = /k + (2,1) exp[+i(p-x)] d 4 x = i[l/(^ -m)] This is the result which was sought. Actually this transformation could have been obtained in an elegant man- ner. For K(2,l) is the Green's function of {\p -m), that is, (i?-m)K(2,l) =16(2,1) (17-1) and it is known that i'p is $ in momentum representation and 6(2,1) is unity. PROBLEMS IN QUANTUM ELECTRODYNAMICS 87 Therefore the transform of this equation can be written down immedi- ately: ($ - m)k(p) = i or k(p) =i/(]rf-m) (17-2) as before. The fact that Eq. (17-1) for K(2,l) has more than one solution is re- flected in Eq. (17-2) in the fact that ($ - mp 1 is singular if p 2 = m 2 . We shall have to say just how we are to handle poles arising from this source in integrals. The rule that selects the particular form we want is that m be considered as having an infinitesimal negative imaginary part. Eighteenth Lecture MOMENTUM REPRESENTATION Since the propagation kernel for a free particle is so simply expressed in momentum representation, k(p) = U(i - m) it will be convenient to convert all our equations to this representation. It is especially useful for problems involving free, fast, moving particles. This requires four-dimensional Fourier transforms. To convert the potential, define ^(q) = /^(x)exp(iq-x) d 4 x (18-1) Then the inverse transform is $(x) = (1/2tt) 4 /a'(q) exp(-iq- x) d 4 q (18-2) The function a(q) is interpreted as the amplitude that the potential con- tains the momentum (q). As an example, consider the Coulomb potential, given by A = 0, cp = Ze/r. Substituting into Eq. (18-1) gives tf(q) =47rZe/(Q-Q)6(q 4 )y t Here the vector Q is the space part of the momentum. The delta func- tion 6(q 4 ) arises from the time dependence of $(x). 88 QUANTUM ELECTRODYNAMICS Matrix Elements. An advantage of momentum representation is the sim- plicity of computing matrix elements. Recall that in space representation the first-order perturbation matrix element is given by the integral M = -i/i(2)e$(2)f(l)d<r 2 For the free particle, this becomes M = -i J u 2 exp(ip 2 •x 2 )e^(2)u 1 expC-ipj • Xj) dT 2 (18-3) In momentum representation, this is simply M = i(u 2 e^(q)u 1 ) (18-3') where ^ i s defined analogously to the three-vector q, The second-order matrix element in space representation is given by - //g(2)e#(2)K + (2,l)e#(l)f(l) dr { dr 2 Substituting for a free particle and also expressing the potential functions as their Fourier transforms by means of Eq. (18-2), this becomes -ffff ^2 exp(ip 2 -x 2 )e^(q 2 ) exp(-iq 2 -x 2 )K + (2,l)e^(qi) x exp (-iqi • x{} u t exp (-ipi • xf)dTj dr 2 • d 4 q 1 /(27r) 4 x d 4 q 2 /(27r) 4 (18-4) If Eq. (18-2) is used for K + (2,l), this kernel can be written K + (2,l) = /i/(jj-m) exp[-ip.(x 2 - Xl )] d 4 p/(27r) 4 Writing the factors that depend on T it this part of the integral is J exp (ip • xj) exp (— iq t • x t ) exp (-ipj • x f ) dTj = (27r) 4 6 4 (p- qi - Pl ) (18-5) where the function 6 4 (x) is to be interpreted as 6(t 1 )6(x 2 )6(y 3 )6(z 4 ). Then the integral over Tj is zero for all $ except $ = $i + 4i« So the integral over p reduces Eq. (18-4) to PROBLEMS IN QUANTUM ELECTRODYNAMICS 89 -ffff U2 ex P (iP2 ' x 2 )e^(q 2 ) exp (-ip 2 • x 2 ) exp [-i(p! + q t ) ■ x 2 ] xi(^i + /4i-mrW(qi)Ui dT 2 d 4 qi /(27r) 4 d 4 q 2 /(27r) 4 Integrating over t 2 results in another 6 function [similar to Eq. (18-5)], which differs from zero only when ^2 - hi = i\ + rii Then integrating over d 4 q 2 gives finally (-i 2 )i / u 2 e^(q 2 )tfi + jzfi -mr'etfq^U! d 4 qi /(27r) 4 (18-6) These results can be written down immediately by inspection of a diagram of the interaction (see Fig. 18-1). The electron enters the region at 1 with FIG. 18-1 wave function Uj and moves from 1 to 3 as a free particle of momentum jz^. At point 3, it is scattered by a photon of momentum 4i [under the action of the potential -ie^(q 1 )]. Having absorbed the momentum of the photon it then moves from 3 to 4 as a free particle of momentum ^ + j4i by conservation of momentum. At point 4, it is scattered by a second photon of momentum 4 2 [under the action of the potential -ie^(q 2 ) absorbing the additional momen- 90 QUANTUM ELECTRODYNAMICS turn # 2 )] • Finally, it moves from 4 to 2 as a free particle with wave func- tion u 2 and momentum $ 2 = $i + A\ + Ai- It is also clear from the diagram that the integral need be taken over q t only, because when jij and $ 2 are given, fa is determined by fa = j^ 2 -^i — fa- The law of conservation of en- ergy requires pj 2 = m 2 , p 2 2 = m 2 ; but, since the intermediate state is a vir- tual state, it is not necessary that (^ + fa) 2 = m 2 . Since the operator V($i + fa _m ) m ay be resolved as (fa + fa + m)/[(^ 1 + j^) 2 — m 2 ], the impor- tance of a virtual state is inversely proportional to the degree to which the conservation law is violated. The results given in Eqs.(18-3') and (18-6) may be summarized by the following list of handy rulest for computing the matrix element M = (u 2 Nuj): 1. An electron in a virtual state of momentum $ contributes the ampli- tude i/($ - m) to N. 2. A potential containing the momentum q contributes the amplitude -iea'(q) to N. 3. All indeterminate momenta q { are summed over d 4 q i /(27r) 4 . Remember, in computing the integral, the value of the integral is desired, with the path of integration passing the singularities in a definite manner. Thus replace m by m - ie in the integrand; then in the solution take the limit as e -* 0. For relativistic work, only a few terms in the perturbation series are necessary for computation. To assume that fast electrons (and positrons) interact with a potential only once (Born approximation) is often sufficiently accurate. After the matrix element is determined, the probability of transition per second is given by P = 27r/(n N)|M| 2 x (density of final states) where II N is the normalization factor defined in Lecture 16. tSee Summary of numerical factors for transition probabilities, R. P. Feynman, An Operator Calculus, Phys. Rev., 84, 123 (1951); included in this volume. Relativistic Treatment of the Interaction of Particles with Light Nineteenth Lecture In Lecture 2 the rules governing nonrelativistic interaction of particles with light were given. The rules stated what potentials were to be used in the calculation of transition probabilities by perturbation theory. Those po- tentials are also applicable to the relativistic theory if the matrix elements are computed as described in Lecture 18. For absorption of a photon, the potential used in nonrelativistic theory was A M =(47re 2 ) 1/2 (2o))- 1/2 e M exp(ik-x) \ K-K -0 (19-1) For emission of a photon, the complex conjugate of this expression is used. These potentials are normalized to one photon per cubic centimeter and hence the normalization is not invariant under Lorentz transformations. In a manner similar to that for the normalization of electron wave functions, photon potentials will, in the future, be normalized to 2a; photons per cubic centimeter by dropping the (2u>)~ 1/2 factor in Eq. (19-1), giving A^ =(47re 2 ) 1/2 e M exp(ik-x) (19-1') This makes any matrix element computed with these potentials invariant, but to obtain the correct transition probability in a given coordinate system, it is necessary to reinsert a factor (2a;)~ 1 for each photon in the initial and final states. This becomes part of the normalization factor IIN, which con- tains a similar factor for each electron in the initial and final states. 91 92 QUANTUM ELECTRODYNAMICS In momentum representation, the amplitude to absorb (emit) a photon of polarization e^ is -i(47re 2 ) e\ The polarization vector e„ is a unit vector perpendicular to the propagation vector. Hence e • e = -1 and e • q = 0. RADIATION FROM ATOMS The transition probability per second is Trans, prob./sec = 27r |H| 2 x (density of final states) where H is the matrix element of the relativistic Hamiltonian, H=a-(-iV-eA) S.R. between initial and final states. That is, <f|H|i> = (47re 2 ) 1/2 /*f*(a -eexp (ik -x)] ^ d vol (19-2) Problem: Show that in the nonrelativistic limit, Eq. (19-2) reduces to l/2m J tf f *[e -p exp(ik-x) +exp(ik- x)p «e +e-(axk) x exp(ik-x)] ^! d vol This is the same result as was obtained from the Pauli equation. SCATTERING OF GAMMA RAYS BY ATOMIC ELECTRONS A relativistic treatment of scattering of photons from electrons will now be given. As an approximation, consider the electrons to be free (energies at which a relativistic treatment is necessary are, generally, much greater than atomic binding energies). This will lead to the Klein-Nishina formula for the Compton -effect cross section. photon 2 (outgoing) photon 1 (incoming >► x recoil electron FIG. 19-1 INTERACTION OF PARTICLES WITH LIGHT 93 For the incoming photon take as a potential A ljU = e ifl exp(-iq 1 -x) and for the outgoing photon take A 2)J = e 2jJ exp (-iq 2 - x). The light is polarized per- pendicular to the direction of propagation (see Fig. 19-1). Thus, ei • qi =0 e 2 -q 2 = also Qi ' <li =c h 2 = and q 2 • q 2 = q 2 2 = (1 9 "3) As initial and final state electron wave functions, choose ^ = uj exp(-ip!-x) * 2 = u 2 exp(-ip 2 -x) where u lf u 2> p t , and p 2 satisfy I^jUj = muj j6 2 u 2 = mu 2 Pi-Pt^m 2 p 2 -p 2 = m 2 (19-4) Conservation of energy and momentum (four equations) is written I*l+rii =^2+^2 (19-5) If the coordinate system is chosen so that electron number 1 is at rest, rfi = my t (19-6a) tf 2 = E 2 y t - p 2 cos 0y x + p 2 sin 0y y (19-6b) ^i=^i(r t ~y x ) (i9-6c) Ai = w 2 (y t - Tx cos e - y y sin e ) (i9-6d) The latter two equations follow from the fact that, for a photon, the energy and momentum are both equal to the frequency (in units in which c = 1). The momentum has been resolved into components. The incoming photon beam can be resolved into two types of polarization, which will be designated type A and type B: (A) rf t = y z (B) i x = y. Type A has the electric vector in the z direction and type B has the elec- tric vector in the y direction. Similarly the outgoing photon beam can be resolved into two types of polarization: 94 QUANTUM ELECTRODYNAMICS (A') i t = y z (B') i 2 = 7 y cos - y x sin 6 Conservation of energy of momentum dictates that either the angle of the recoil electron <fi or the angle at which the scattered photon comes off completely determines the remaining quantities. If the electron direction is unimportant, its momentum can be eliminated by solving Eq. (19-5) for fa and squaring the resulting equation: fa = fa + A\ ~ Ai p 2 2 = m 2 = (fa + fa - fa)(fa + fa - 4 2 ) = Pi 2 + Qi 2 + Q2 2 + 2pi ■ q t - 2 Pl • q 2 - 2q t • q 2 = m 2 +0+0 +2mco 1 -2maJ 2 -2cL) 1 a;2 (1 - cos 8) where the last line was obtained from the preceding line by using Eqs. (19-3), (19-4), and (19-6a, c, d). This can be written m(u) 1 - u) 2 ) = co 1 a; 2 (l -cos 6) or (m/a; 2 ) - (m/wj) = 1 - cos 9 (19-7) This is the well-known formula for the Compton shift in wavelength (or fre- quency) . DIGRESSION ON THE DENSITY OF FINAL STATES By the method discussed in the earlier part of the course, the following final state densities (per unit energy interval) can be obtained. When a sys- tem of total energy E and total linear momentum p disintegrates into a two- particle final state, Pl 3 d Q t Density of states = (27r)" J E 1 E 2 — — ~ 2 — (D-l) E Pl ~ E i(P - Pi) where E t = energy of particle 1; E 2 = energy of particle 2; pj = momentum of particle 1; dfij = solid angle, into which particle 1 comes out; m t = mass of particle 1; m 2 = mass of particle 2; and Ej + E 2 = E, pi + p 2 = p. Another useful formula is in terms of the final energy of particle 1 and its azimuth t (instead of it 0j). It is Density of states = (27r)" 3 (E t E 2 / |p|) dEj d<f> t (D-2) INTERACTION OF PARTICLES WITH LIGHT 95 Special cases: (a) When m 2 = °° (E 2 = °°, E = °°): Density of states = (2tt)- 3 EjpJ dfij (D-3) (b) In center-of-mass system p = 0: Density of states = (2tt)- 3 [E 1 E 2 d^/^ +E 2 )] (D-4) When a system disintegrates into a three-particle final state, P 2 3 Pi 2 dp t dfij dfi 2 Density of states = (2*)-* E 3 E 2 ^ - _ - _ — ; ^^ (D-5) Special case: When m 3 = °°: Density of states = (27r)- 6 E 2 |p 2 | dfi 2Pl 2 dp t dSl t (D-6) The Compton effect has a two-particle final state: taking particle 1 to be photon 2 and particle 2 to be electron 2, from Eq. (D-l), w 2 3 dfi w Density of states = (27T)"" 5 o^E? -; : — 2 ; 7^ (m + <jOi)<jo 2 - w 2 (a; 1 a; 2 cos 6) COMPTON RADIATION Calculation of |M| 2 . Using the Compton relation Eq. (19-7) to eliminate 6, this becomes Density of states = (27r) -3 (E 2 w 2 3 dn o ,/ma) 1 ) The probability of transition per second is given by Trans, prob./sec =<jc= (27r/2E 1 2E 2 2a; 1 2a; 2 ) |M| 2 x (27r)- 3 (E 2 a;2 3 dft /mc^) or a = [w 2 2 dfi ( ./(27r) 2 16m 2 w 1 2 ] |M| 2 In working out the matrix element M, there are two ways in which the scat- tering can happen: (R) the incoming photon is absorbed by the electron and then the electron emits the outgoing photon; (S) the electron emits a photon and subsequently absorbs the incident photon. These two processes are shown diagrammatically in Fig. 19-2. 96 QUANTUM ELECTRODYNAMICS In momentum representation, the matrix element M for the first proc- ess R is i[-i(47r e 2 ) 1/2 ] 2 { u 2 ^ 2 [l/O^ + fl! - m)] i x uj Reading from right to left the factors in the matrix element are interpreted as follows: (a) The initial electron enters with amplitude Uj; (b) the elec- tron is first scattered by a potential (i.e., absorbs a photon); (c) having re- FIG. 19-2 ceived momentum ^ from the potential the electron travels as a free elec- tron with momentum $ t + jz(i; (d) the electron emits a photon of polarization ^ 2 ; and (e) we now ask for the amplitude, that the electron is in a state u 2 . Exercise: Write down the matrix element for the second process S. The total matrix element is the sum of these two. Rationalize these matrix elements and, using the table of matrix elements (Table 13-1) work out |M| 2 . Twentieth Lecture For the R diagram, M was found to be -i47re 2 {u 2 ^ 2 [V(^i + tfi - m)]^^} = -i47r e 2 ( u 2 Ru 1 ) and as an exercise the matrix element for the S diagram was found to be INTERACTION OF PARTICLES WITH LIGHT 97 -i47re 2 { u 2 ^i[l/(^i -ffo-m)]^^} = -i47re 2 ( u^) The complete matrix element is the sum of these, so that the cross section becomes cr = (e 4 /4m)(w 2 2 /wi 2 ) dfi 2 I u 2 (R + S)u 1 | 2 The problem now is actually to compute the matrix elements for R and S. First R will be considered. Using the identity 1/(1* -m) =(]6 + m)/(p 2 -m 2 ) the matrices may be removed from the denominator of R giving (#1 +ii) - m 2mco! The denominator is seen to be 2mo) 1 from the following relations: (rfl + til) 2 - m 2 = p! 2 + 2 Pl • q t + qj 2 - m 2 Pi = m qi 2 = 2 Pi *qi = 2mw The matrix elements for the various spin and polarization combinations can be calculated straightforwardly from this point. But certain preliminary manipulations will reduce the labor involved. Using the identity M =2a-b - Vi it is seen that ^2^i^i = ^2(2pre 1 ) - M\$x But Pt has only a time component and e t only a space component so Pi •e 1 = 0. Recalling that ^uj = mui, it is seen that u 2 ^ 2 ^i^iui = -u 2 ^2^i#iui = -(u 2 ^ 2 ^ 1 u 1 )m and this is the matrix element of the first term of R. It is also the negative of the matrix element of the last term of R, so R may be replaced by the equivalent R =^ 2 ^i^i/2mo; 1 98 QUANTUM ELECTRODYNAMICS By an exactly similar manipulation, the S matrix is equivalent to Substituting fa = u>i(y t -y x ) and fa - w 2 (y t ~ ?x cos ^ ~ T y sin 0) and trans- posing the 2m factor, the complete matrix may be written 2m(R + S) = fa(y t -y % )fa+fa(y t -y x cos 9 -y y sin 9) fa A still more useful form is obtained by noting that e^ anticommutes with Qitei'Qi = °) and ii witn ^2 and that iii\ = 2e2-e! - ^i^ 2 . Thus, 2m(R + S) = -fa 4 x (y t - y x ) - fa fa(y t - y x + y x - y x cos - y y sin 0) = -2(e 2 • ei)(y t - y x ) - fa fa [y x (l - cos 0) - y y sin 0] Using this form of the matrix, the matrix elements may be computed easily. For example, consider the case for polarization: fa -y t , 4i~ 7 y cos 9 ~ y x sin 9. This corresponds to cases (A) and (B') of Lecture 19 and will be de- noted by (AB'). The matrix is 2m(R + S) = -y z (y y cos 9 - y x sin 0)[y x (l - cos 9) - y y sin 0] since e 2 »ei =0. Expanded this becomes 2m(R + S) = -y z [y y y x cos 0(1 cos 0) + cos sin0 + sin 9(1 -cos 9) + y x y y sin 2 9) = -y z (y x y y ~ yx^cos 9 + sin 9) = -y x y y y z (l - cos 9) - y z sin 9 where the anticommutation of the y 's has been used. In the case of spin- up for the incoming particle and spin down for the outgoing particle (s t = -1), s 2 = -1), the matrix elements -2m (FiF^ 172 ( u 2 y x y y y z Ul ) = -iF 2 p 1+ - iFj p 2 + -2m(F 1 F 2 ) 1/2 (u 2 y z u 1 ) = +p t + F 2 -p 2 + F t - may be found by reference to Table 13-1. But note that in this problem p t + = p xl + ip yl = since particle 1 is at rest. Hence the final matrix element for this case, polarization (AB 7 ), spin s A = +1, s 2 = -1, is >* ^ >t •^ ^jl I o o X I + Q5 o7 •iH CQ CO o <D> X o rt 1 1 CO CD N c^ CO o o X 1 »f 1 c*- + •Hi 05 rH h I CM a as CQ O o I CM P>T CD •t-i CQ + CM a + pC CM a Q5 Ph CQ O CE> O fl 1 1 CQ pC + CM a? P*T CQ o o Q5 a 1 CQ tH 1 ( t n(g+H) z ^) z J I JA ul g CQ a> -»-> CTJ hD 3 •i— > PS o o X CD J2 > £ o o O ^2 a y. . r-l tH <D + 1 xi +j II II a> 1-1 CNJ Sh CQ CQ aS H iH U O 1 1 «M II II CD CQ i-l CM CQ CQ O +-> <4-l U o O <4H CQ CQ B PI CD £ hfl "5* o <D o X X <D 'u -t-> 'a £ £ o CD o J3 CD H -t-> oi CD o cd ^ >> CD ,fl 100 QUANTUM ELECTRODYNAMICS 2m(F 1 F 2 ) 1/2 (u 2 (R + S)u 1 ) =-(1 - cos 9)iF i p 2 + - sin 9 P 2+ F t The results for the other combinations of polarization and spin are obtained in the same manner and will only be presented in tabular form (Table 20-1). They may be verified as an exercise. For any one of the polarization cases listed, |M| 2 is the sum of the square amplitudes of the matrix elements for outgoing spin states averaged over in- coming spin states. But this is seen to be simply the square magnitude of the nonzero matrix element listed under the appropriate polarization case. For example, in case (AA'), |M| 2 = |u 2 (R + S) Ul | 2 = (l/4m 2 F 1 F 2 ) ^F^ - (1 +cos 0) F lP2 _ - i sin 9 FiP 2 + By employing the relation p 2 _ = Pi_ + qt_ - q 2 _ = qt_ - q 2 _ = U\ - oo 2 cos 9 + ioj 2 sin 9 and (m/o; 2 ) — (m/a^) = 1 - cos 9 the square magnitudes of the matrix elements for the various cases reduce, after considerable amount of algebra, to the expressions given in Table 20-2. TABLE 20-2 Polarization |M| 2 AA' [(a)! - u> 2 ) 2 /u) jo; 2 J + 4 AB' [(ct)t -co 2 ) 2 /o; 1 ct) 2 ] BA X [(W! -co 2 ) 2 /o; 1 u) 2 J BB 7 [(ct)! -w 2 ) 2 /o; 1 a) 2 ] + 4 cos 2 9 It is clear that all four of these formulas may be written simultaneously in the form |M| 2 = [(wt - co 2 ) 2 /o; 1 a) 2 ] + 4(e t ■ e 2 ) 2 Note that these formulas are "not adequate for circular polarization. That is, if ^! were, for example, 1/V2 - (iy z + y y ), it is seen that because of the phas- INTERACTION OF PARTICLES WITH LIGHT 101 ing represented by the imaginary part of ^ 1? all the calculations must be carried out before squaring the matrix elements in order to get the proper interference. Finally the cross section for scattering with prescribed plane polariza- tion of the incoming and outgoing photons is a = (e 4 /4m 2 )(a; 2 2 /co 1 2 ) dftc^ [(w 2 /^i) + (^/w 2 ) ~ 2 + 4( ei • e 2 ) 2 ] This is the Klein-Nishina formula for polarized light. For unpolarized light this cross section must be averaged over all polarizations. It is noted that diagram cases such as Fig. 20-1 have been included in di &i FIG. 20-1 FIG. 20-2 the previous derivation as a result of the generality in the transformation of of K + (2,l) to momentum representation. In fact, all diagram cases have been included except higher-order effects to be discussed later. (They corre- spond to emission and reabsorption of a third photon by the electron, such as in Fig. 20-2.) Twenty -first Lecture Discussion of the Klein-Nishina Formula. In the "Thompson limit," U3\ «m. Tnen the electron picks up very little energy in recoil, and c^^u^- This can be seen from the relation mwi - mct) 2 = k>i w 2 (1 - cos 6) In this limit, the Klein-Nishina formula gives cr=(e 4 /m 2 )(e 1 -e 2 ) 2 dn aj (21-1) (21-2) 102 QUANTUM ELECTRODYNAMICS which is the Rayleigh-Thompson scattering cross section. Note that w is still very large compared to the eigenvalues of an atom, in accordance with our original assumptions for Compton scattering. The same result is obtained by a classical picture. Under the action of the electric field of the photon E = E e! exp (icot), the electron is given the acceleration a = (e/m)E e t exp (iwt) Classically, an accelerated charge radiates to give the scattered radia- tion _ __§_ (retarded acceleration projected on plane 1 to s ~ R line of sight) The scattered radiation polarized in the direction e 2 is determined by the component of the acceleration in this direction. The intensity of the scat- tered radiation of polarization e 2 is then (times R 2 per unit solid angle and per unit incident intensity) I=(e 4 /m 2 )(e r e 2 ) 2 (21-2') The customary fi 's and c's may be replaced in Eq. (21-1) as follows (cr is an area or length squared): e 4 = (e 2 ) 2 = (e 2 /Rc) 2 m 2 = (mc/K) 2 = length squared e 4 /m 2 = (e 2 /mc 2 ) 2 = r 2 » 8 x 10 -26 cm 2 Averaging over Polarizations. It is often desired to have the scattering cross section for a beam regardless of the incoming or outgoing polariza- tion. This can be obtained by summing the probabilities over the polariza- tions of tne outgoing beam and averaging over the incoming beam. Thus, suppose the incoming beam has polarization of type A. The probabilities (or cross sections) for the two possible types of outgoing polarization, A' and B' can be symbolized as AA' and AB'. The total probability for scat- tering a photon of either polarization is AA' + AB' . Then suppose the incom- ing beam is equally likely to be polarized as type A or type B. The result- ing probability can be obtained as the sum 1/2 (probability if type A) + 1/2 (probability if type B). This is the situation for unpolarized incoming beam, and gives a (averaged over = U/2XAA' + AB') + (1/2)(BA' + BB') polarizations) 4N d« (0* + OH- sir? e) (21-3) INTERACTION OF PARTICLES WITH LIGHT 103 If, on the other hand, the polarization of the outgoing beam is measured (still with an unpolarized incoming beam), its dependence on frequency and scattering angle is given by the ratio Probability of p ola rization type A! _ (1/2)[AA / + BA'] Probability of polarization type B' ~ (1/2)[AB' + BB'] (fcjg/gj) + (^1/^2) (w 2 /wi) + (w 1 /a;2)-2sin 2 9 The forward radiation (0 = 0) remains unpolarized, but a certain degree of polarization will be found in light scattered through any nonzero angle. In the low-frequency limit (wi^u^), the polarization is complete at 9- n/2. Thus an unpolarized beam becomes plane -polarized when scattered through 90°. t Total Scattering Cross Section. If the cross section (averaged over polar- izations) given in Eq. (21-3) is integrated over the solid angle dfi = 27r d(cos 9) = (27rm/a; 2 2 ) da> 2 the total cross section for scattering through any angle is obtained. So, from Eq. (21-1), cos 9=1- m/u 2 + m/u>i (2 1-1') and the variable a> 2 goes between the limits mcoi/(2wi + m) and uj^ as cos 9 goes from - 1 to + 1. Equation (21-3) can be written da T = (e 4 /2m 2 )(27r/o) 1 2 )m d W2 (co 2 /wi + coi/co 2 - 2m/co 2 + 2m/cj 1 + m 2 /o)i 2 + m 2 /o) 2 2 - 2m 2 /u>ico 2 ) where the last five terms replace -sin 2 6 = cos 2 9 - 1 using Eq. (2 1-1'). Simple integrations yield? o- T = 7re 4 /m 2 [(m/uj 1 -2m 2 /wi 2 -2m 3 /wi 3 ) log(2w 1 /m+l) + m/2u>i + 4m 2 /wi 2 -m 3 /2w 1 (2w 1 + m) 2 ] In the high-frequency limit (wi"* 00 ) o" T ~ (l/c^i) log u)\ -* t Cf. Walter Heitler, "Quantum Theory of Radiation," 3rd ed., Oxford, 1954; and B. Rossi and K. Greissen, Phys. Rev., 61, 121 (1942). % Cf. Heitler, op. cit., p. 53. 104 QUANTUM ELECTRODYNAMICS Thus Compton scattering is a negligible effect at high frequencies, where pair production becomes the important effect. TWO-PHOTON PAIR ANNIHILATION From the quantum -electrodynamical point of view, another phenomenon completely analogous to Compton scattering is two-photon pair annihilation. Two photons are necessary (in the outgoing radiation) to maintain conser- vation of momentum and energy when pair annihilation takes place in the absence of an external potential. The interaction can be diagrammed as shown in Fig. 21-1. This figure should be compared to that for Compton scattering (Lecture 20). The only differences are that the direction of pho- ton fa is reversed, and, since particle 2 is a positron, ^ 2 = -(momentum of positron). So write i x = (E_y t -p. y) ii = -(E+Yt -p + *y) ii zjj FIG. 21-1 where the energies E_ and E + of the electron and positron are both posi- tive numbers. The conservation law gives $i = i\~ fa- fa (21-4) (just as for Compton scattering, but the direction of fa reversed), so the matrix element for this interaction is M t = -i47re 2 (u 2 e / 2 (jfj -fa- m)- 1 ^^) The second possibility, indistinguishable from the first by any measure- INTERACTION OF PARTICLES WITH LIGHT 105 ment, is obtained from the first by interchanging the two photons (see Fig. 21-2); again note similarity to Compton scattering. Immediately, the matrix element is M 2 = -i 4?re 2 ( u 2 4\ (& - ii ~ m) -1 & 2 "i) ii - At FIG. 21-2 The sum of the two matrix elements and the density of final states gives the cross section cr • (velocity of positron) = 27r/(2E_ • 2E + • 2u 1 ' 2lo 2 ) • iMj + Mgl 2 x (density of states) in a system where the electron is at rest and the positron is moving. The density of final states is oj 1 o;2/(27r) 3 co* d^ 1 /(w 2 w 1 - Q2*Qi) Since particle 2 is a positron, $ 2 = ~i>+, so the conservation law, Eq. (21-4), gives & + tf+ = i\ + ii Then m 2 + 2(pj • p + ) + m 2 = + 2qt ■ q 2 + This reduces to 2nr + 2mE + = 2w 1 oj 2 - 2Qr 106 QUANTUM ELECTRODYNAMICS Taking the velocity of the positron as |p + |/E + , the cross section is or= (27r)o; 1 2 dn 1 /[2E^-2|p + |4(27r) 3 -m(E + + m)] x|M! + M 2 | 2 ufdttj |Mj + M 2 [ 2 ~64tt 2 m 2 |p + | (m + E + ) From a comparison of the diagrams, it is clear that the matrix elements for pair annihilation are the same as the matrix elements for the Compton effect if the sign of sk\ is changed. In the cross section, this amounts to changing the sign of a^. Then the cross section is a =e 4 cV dfi 1 /[4m 2 (E + + m)|p + |][(w 2 /a; 1 )+(a; 1 /a; 2 ) + 2 -4(e-e 2 ) 2 ] in analogy with the Klein-Nishina formula. Twenty -second Lecture POSITRON ANNIHILATION FROM REST The formula for positron-electron annihilation derived in Lecture 21 di- verges as the positron velocity approaches zero (a ~ 1/v; this is true for other cross sections when a process involves absorption of the incoming particle, and is the well-known 1/v law). To calculate the positron lifetime in an electron density p (recall that the preceding cross section was for an electron density of one per cubic centimeter) as v + — - 0, we use Trans, prob./sec = crv+p plus the fact that, as v + — - 0, E + — - m and u\ ~ * w 2 ~* m (when the electron and positron are both approximately at rest, momentum and energy can be conserved only with two photons of momenta equal in magnitude but opposite in direction). Thus Trans, prob./sec =crv + p= (e 4 /2m 2 )pdft (sin 2 0) (22-1) where 6 = angle between directions of polarization of two photons (cos 9 = e 1 -e 2 ). The sin 2 6 dependence indicates that the two photons have their polarizations at right angles. To get the probability of transition per second for any photon direction and any polarization, it is necessary to sum over solid angle (jdQ,= 47r) and average over polarizations (sin 2 6 = 1/2), giving Total trans, prob./sec = 1/t = (7re 4 /m 2 )p = 7r(e 2 /mc 2 ) 2 cp = 7rr 2 cp (22-2) INTERACTION OF PARTICLES WITH LIGHT 107 (factors of c and H reinserted where required), where r = classical elec- tron radius, and t = mean lifetime. Problems: (1) Obtain the preceding result directly by using matrix elements for an electron and positron at rest. Show that only the sin- glet state (spins antiparallel) can disintegrate into two photons. The triplet state disintegrates into three photons and has a longer lifetime (see the next problem). (2) Find the mean time required for a positron and electron to dis- integrate into three photons (spins must be parallel). The following procedure is suggested: (1) set up formula for rate of disintegration; (2) write M in the simplest possible form; (3) make a table of matrix elements (same as Table 13-1 but with fa - my t , fa ~ — rny t ); (4) find the matrix element of M for eight polarization cases; (5) find the rate of disintegration for each case; (6) sum the disintegration rate over polarizations; (7) obtain the photon spectrum; (8) obtain the total dis- integration rate by integrating over photon spectrum and angle; and (9) compare with Orr and Powel.t (3) It is known that the matrix elements should be independent of a gauge transformation $ - £ + & <&> where a is an arbitrary constant and (;{ is the momentum of a photon whose polarization is e' or e 7 '. Show that substituting sk f° r ^ m the matrix elements for the Comp- ton effect gives m - 0. BREMSSTRAHLUNG When an electron passes through the Coulomb field of a nucleus it is de- flected. Associated with this deflection is an acceleration which, according to the classical theory, results in radiation. According to quantum electro- dynamics, there is a certain probability that the incident electron will make a transition to a different electron state with a photon emitted, while in the field of the nucleus. Interaction with the field of the nucleus is necessary to satisfy conservation of energy and momentum. That is, the electron cannot emit a photon and make a transition to a different electron state while trav- eling along in a vacuum. Figure 22-1 shows the process and defines angles that arise later. The Coulomb potential of the nucleus will be considered to act only once (Born approximation). The validity of this approximation was discussed in Lecture 16. There are two (indistinguishable) orders in which the brems- strahlung process can occur: (a) the electron interacts with the Coulomb field and subsequently emits a photon, or (b) the electron first emits a pho- ton and then interacts with the Coulomb field. The diagrams for these proc- tA. Ore and J. L. Powell, Phys. Rev., 75, 1696 (1949). 108 QUANTUM ELECTRODYNAMICS esses are shown in Fig. 22-2. The interaction with the nucleus gives mo- mentum to the electron. Conservation of energy and momentum requires i\ + $ = A + 4 or = ii - i\ + i electron 1 ► Coulomb field of the nucleus FIG. 22-1 electron 2 photon f(Q) V(Q) — > i\~<A=fa + 4 (a) FIG. 22-2 In Lecture 18 it was shown that the Fourier transform of. the Coulomb poten- tial was proportional to 6(Q/\), since the potential is independent of time. This means that only transitions for which Q 4 = occur, or energy must be conserved among the incident electron, final electron, and photon. Thus Ej = E 2 + w. The transition probability is given by Trans, prob./sec = gv^ = (27r/2E 1 2E 2 2u;) |37l| 2 xD INTERACTION OF PARTICLES WITH LIGHT 109 Since the nucleus is to be considered infinitely heavy, D = (2tt)- 6 E 2 p 2 dfi 2 w 2 dw dft w Notice that there is a spectrum of photons; that is, the photon energy is not determined (as it was in the Compton effect, for example). Letting SHT = (UjMU!), M = (-i)<4re*)* [* ~~z^ m) + m) -~ p 2 - Q - m (22-3) where the first term comes from Fig. 22-2a and the second term from Fig. 22 -2b. The explanation of the factors in the first term, for example, is, reading from right to left, that an electron initially in state Uj is scattered by the Coulomb potential acquiring an additional momentum $ , the electron moves as a free particle with momentum j^ + $ until it emits a photon of polarization e 7 . We then ask: Is the electron in state u 2 ? For the Coulomb potential >f(Q) = (47rZe 2 /Q 2 )6(Q 4 )y t = v(Q)6(Q 4 )Yt (see Momentum Representation, Lecture 18) in a coordinate system in which the nucleus does not move. [For potential other than Coulomb, use appro- priate v(Q), the Fourier transform of the space dependence of the potential.] Rationalizing the denominator of the matrix,! M = (-i)(47re 2 ) l/2 v(Q) i\ + $ + m [2 y% + Y t L" -2p 1 -Q.-Q - i 2 - Q + m 2p 2 • Q - Q 2 P _ (22 -4) The outgoing photon can be polarized in either of two directions, and the in- coming and outgoing electron each have two possible spin states. The vari- ous matrix elements can be worked out using Table 13-1 exactly as was done in deriving the Klein-Nishina cross section in Lecture 20. Nothing new is involved, so we omit the details. After (1) summing over photon polari- zations, (2) summing over outgoing electron spin states, and (3) averaging over incoming electron spin states, the following differential cross section is obtained: t(rfi +&- m)(^i + + m) = p t 2 + 2 Pl • Q + Q 2 - m 2 = 2 Pl • Q + Q 2 = -2 Pl • Q + Q 2 = 2pi • Q - Q 2 Q 4 = o 110 QUANTUM ELECTRODYNAMICS 1 /Ze 2 \ 2 2 dw P2 . Q ' a . fl , fl ,, da = — -I — o- e l sin 2 d# 2 sin 6< d8 t d<2> 27T \ Q / W Pj x f p 2 2 sin 2 2 (4E 1 2 -Q 2 ) + p^sin 2 ^ (4E 2 2 -Q 2 ) 1 (E 2 - p 2 cos 2 ) 2 (Ej - pt cos 0i) 2 2p t p 2 sin^t sin# 2 cos (p (4E t E 2 -Q 2 + 2o; 2 ) -2a; 2 (p 2 2 sin 2 fl 2 + p t 2 sin^) (E 2 - p 2 cos 2 )(E 1 - pj cos t ) (22-5) An approximate expression with a simple interpretation in terms of the Coulomb elastic scattering cross section can be obtained when the photon energy is small (small compared to rest mass of electron but large com- pared to electron binding energies). Writing the matrix (22-3) in terms of gf instead of , m = (-imW)^ [i ^^ y<Q) + m ^^ i ] + m 1 = (-i)(47re 2 ) l/2 |rf +2v yW) + ^(Q) ■2p using the relationships ^ 2 = -tf 2 & + 2e • p 2 , jz^e 7 = —4$\ + 2e • p lf and neglecting g[ in the numerator, since it is small, this becomes M«(-i)(47re 2 ) 1/2 v(Q) •g^Tt + 2e ; p 2 y t + m^y t 2p 2 -q -Vt&h + 2prey t + me / y t (-i)(47re 2 ) l/2 v(Q) 2prq e-p t ep 2 6(Q 4 ) q*Pi q-P2 r t 6(Q 4 ) where use is made of the fact that the matrix element of M between states u 2 and u t is to be calculated and u 2 ^ 2 = u 2 m, ^Uj = muj. The cross section for photon emission can then be written dcr = 2E!2E 2 ' ™ }l (2tt) 3 e 2 do; • dQ CL , /p 2 • e p t ♦ e \ 2 7TCt) q q v P2 *^ Pi ^; The first bracket is the probability of transition for elastic scattering (see Lecture 16), so the last bracket may be interpreted as the probability of photon emission in frequency interval da; and solid angle d£2 w if there is elastic scattering from momentum p t to p 2 . INTERACTION OF PARTICLES WITH LIGHT 111 Problem: Calculate the amplitude for emission of two low-energy photons by the foregoing method. Neglect q's in the numerator but not in the denominator. Ansiver: Another factor, similar to that in the preceding equations, is obtained for the extra photon. PAIR PRODUCTION It is easily shown that a single photon of energy greater than 2m cannot create an electron positron pair without the presence of some other means of conserving momentum and energy. Two photons could get together and create a pair, but the photon density is so low that this process is extremely unlikely. A photon can, however, create a pair with the aid of a field, such as that of a nucleus, to which it can impart some momentum. As with brems- strahlung, there are two indistinguishable ways in which this can happen: (a) The incoming photon creates a pair and subsequently the electron inter- acts with the field of the nucleus; or (b) the photon creates a pair and the positron interacts with the field of the nucleus. The diagrams for these al- ternatives are shown in Fig. 22-3. The arrows in the diagram indicate that nQ) (a) (b) FIG. 22-3 ^! is the positron momentum and ^ 2 * s the electron momentum. Notice that, with respect to the directions that the arrows point (and without regard to direction of increasing time), these diagrams look exactly like those for the bremsstrahlung process: Starting with fa in case (a), the particle is first scattered by the Coulomb potential and then by the photon; in case (b) the order of the events is reversed. The difference between pair production and bremsstrahlung, when the direction of time is taken into account, is (1) fa is a positron state (an electron traveling backward in time), and (2) the photon ^ is absorbed rather than emitted. As a result, the bremsstrahlung matrix elements can be used for this process if fa is replaced by -$ + and ^ by -&. 112 QUANTUM ELECTRODYNAMICS The ^ + is then the positron momentum and <& is the momentum of the ab- sorbed photon. The density of final states is different, of course, since the particles in the final state are now a positron and electron. Thus da= (l/27r)(Ze 2 /Q 2 ) 2 e 2 (p + p_ sin0 + d0 + sin0_ d0_ d<p/u 3 ) x{ } (22-6) where the braces are the same as for bremsstrahlung, Eq. (22-5), except for the following substitutions: P- for p 2 "P+ for Pi -6_ for e 2 -0 + for 6_ E_ for E + -E + for E t -gl> for u Figure 22-4 defines the angles ($ = angle between electron-photon plane and positron-photon plane). positron electron photon FIG. 22-4 Twenty -third Lecture A METHOD OF SUMMING MATRIX ELEMENTS OVER SPIN STATES By using current methods of computing cross sections, one first arrives at a cross section for "polarized" electrons, that is, electrons with definite incoming and outgoing spin states. In practice it is common that the incom- ing beam will be "unpolarized" and the spins of the outgoing particles will be unobserved. In this case, one needs the cross section obtained from that for "polarized" electrons by summing probabilities over final spin states and averaging this sum over initial spin states. This is the correct process since the final spin states do not interfere and there is equal probability of initial spin in either direction. Formally, if (T-U^MU!) INTERACTION OF PARTICLES WITH LIGHT 113 one needs \ Tj Tj l^ 2 Mu 1 )| 2 (23-1) 2 spins 1 spins 2 where Yj me ans the sum over final spin states for only one sign of the spins 2 the energy, that is, over only two of the four possible eigenstates. Similarly, Y) is the sum over initial spins for one sign of the energy. The purpose spins 1 now is to develop a simple method for obtaining these sums . In accordance with the usual rule for matrix multiplication, the following is true: Yj (u 2 Au 1 )(u 1 Bu 2 ) = 2m(u 2 ABu 2 ) (23-2) all u x where A and B are any operators or matrices, the 2m factor on the right arises from the normalization uu = 2m, and the sum is over all eigenstates represented by uj. But the states u, which we want in Eq. (23-1) are not all states, just those satisfying i>\\\\ = mu t . That is, they belong to the eigen- value + m of the operator jfj. Since ^i 2 = m 2 , ^ also has the eigenvalue -m, that is, there are two more solutions of jz^u = -mu which, together with the two we wish in Eq. (23-1) bring the total to four. Let us call the latter " negative eigenvalue" states. Now, if in Eq. (23-2) the matrix elements of B were zero in negative eigenvalue states, this would be the same as £y » that is, just over posi- tive eigenvalue states. So consider spins 1 2} (u 2 Au 1 )(u 1 (^ 1 + m)Bu 2 ) = (u 2 A(^ 1 + m)Bu 2 )2m all u t But u^^+m) = for negative eigenvalue states = Uj(2m) for positive eigenvalue states so the preceding sum also equals X) (u 2 Au 1 )2m(u 1 Bu 2 ) spins 1 Cancelling the 2m factors, this gives 2 (u 2 Au 1 )(u 1 Bu 2 ) = (u 2 A(& + m)Bu 2 ) spins 1 (^! + m) is called a projection operator for obvious reasons. Similarly it follows that 114 QUANTUM ELECTRODYNAMICS £ (u 2 Xu 2 ) = £ <l/2m)(u 2 0fe + m)Xu 2 ) spins 2 all u 2 where X is again any matrix. Remembering the normalization u 2 u 2 - 2m, it is seen that the last sum is just the trace or spur of the matrix $ 2 + m)X. Note that the order of X and $ 2 + m is immaterial. Finally, when one wants 2 2 |<u 2 m Ui )1 2 spins 1 spins 2 collection and specialization of the previous results is seen to give 1/2 £ £ |u 2 MUi| 2 »l/2 £ £ (uaMutM^Muz) spins 1 spins 2 spins 1 spins 2 = 1/2 Sp[(^ 2 + m)M(^ 1 + m)M] (23-3) where the last notation means the spur of the matrix in the brackets. It is true whether $j, $ 2 represent electrons or positrons. The following list of the spurs of several frequently encountered matrices may be verified easily: Sp[l] = 4 Sp[y M ] = Sp[xy] = Sp[yx] Sp[x + y] =Sp[x] + Sp[y] Spiral = ° if M * v = +4 if jit = v = 4 = -4 if jli = v= 1, 2, 3 SpMJfl = 1/2 Sptftf + Kd] = Sp[a • b] = 4 a • b SpLa^ = It is also true that the spur of the product of any odd number of daggered operators is zero. Sp[(^ 1 + m 1 )(^ 2 -m 2 )] = Sp[^ 2 ] + Sp[m^ 2 - ^im 2 - m^] = 4(p 1 -p 2 - m^j) (23-4) Sp[$!+ m!)(^ 2 - m 2 )(tf 3 + m 3 )$4-m 4 )] = 4(p 1 -p 2 -m 1 m 2 )(P3 , P4-m 3 m4) - 4(pj • p 3 - m^g) x (p 2 • p 4 - m 2 m 4 ) + 4(pj • p 4 - m 1 m 4 )(p 2 ■ p 3 - m 2 m 3 ) (23-5) INTERACTION OF PARTICLES WITH LIGHT 115 As an example, the case of Coulomb scattering will be ' 'treated" using this technique. The cross section for polarized electrons was previously found to be cr=(Z 2 e 4 /Q 4 )|(u 2 y t u 1 )| 2 Therefore, since y t = y t , the cross section for unpolarized electrons is, by Eq. (23-3), ^unpoi = 1/2 (Z 2 e 4 /Q 4 ) Sp[(^ 2 + mfrtOfc + m)y t ] The spur can be evaluated immediately from Eq. (23-5) with m 2 = m 4 = and F*2 = $4 = Tt • Another way is: Since y t ^i = 2Ej - $iJ t , it is seen that $ 2 + m)y t (& + m)y t = $ 2 + m)(2E 1 y t - jf t + m) Using a few of the formulas listed previously, the spur of this matrix is seen to be - 4 Pl *P2 + 8E 1 E 2 + 4m2 But Pi • p 2 = E^ - Pi • P2 , Pi* P2 = P 2 cos 9, and Ej = E 2 , so this is 4E 2 + 4m 2 + 4p 2 cos Also m 2 = E 2 — p 2 , so that finally the cross section becomes = 1/2 (Z 2 e 4 /Q 4 )[8E 2 + 4p 2 (cos 9 - 1)] unpol -2«4 /^4x T72 = (4Z^eVQ 4 )EMl - v £ sin' (6/2)] where v 2 = p 2 /E 2 . This is the same cross section obtained previously by other methods. EFFECTS OF SCREENING OF THE COULOMB FIELD IN ATOMS The cross sections for the pair production and bremsstrahlung processes contained the factor [V(Q)] 2 , where V(Q) is the momentum representation of the potential; that is, V(Q) = /v(R) exp (-iQ • R) d 3 R which for a Coulomb potential is V(Q)=47rZe 2 /Q 2 where Q is the momentum transferred to the nucleus or p x - p 2 - q. 116 QUANTUM ELECTRODYNAMICS Clearly V(Q) gets large as Q gets small. The minimum value of Q oc- curs when all three momenta are lined up (Fig. 23-1): Pi P2 lQ min l = pi - p 2 - q = Ipil - I'PsI - ( E i - E 2) For very high energies E »m, E - p « m 2 /2E so that in this case Qmin = (m 2 /2)[(l/E 2 ) - (1/E,)] « mV2EtE 2 From this it is seen that Q min ~* as E| -*• °° . This shows clearly why the cross sections for pair production and bremsstrahlung go up with energy. From the integral expression for V(Q) it is seen that the main contribu- tion to the integral comes when R ~ l/Q. So as Q becomes small the im- portant range of R gets large. It is in this way that screening of the Cou- lomb field becomes effective. The value of 1/Q min for a contemplated proc- ess can be estimated from the foregoing formula. The atomic radius is given roughly by a Z~ 1/3 , where a is the Bohr radius. Thus if Re.f = l/Q min > a„Z- 1/3 or, what is the same, EjEg/q > 1/2 (137) mZ*" l/3 then screening effect will be important, and vice versa for the opposite in- equalities. If from this estimate screening would appear to be important, one should use the screened Coulomb potential. It gives the result V(Q) = (47re 2 /Q 2 )[Z - F(Q)] where F(Q) is the atomic structure factor given by F(Q) = /n(R) exp(-iQ'R) d 3 R and n(R) is the electron density as a function of R. INTERACTION OF PARTICLES WITH LIGHT 117 Twenty -fourth Lecture Problem: In discussing bremsstrahlung it was found that the cross section for emission of a low-energy photon can be approximated as cr = oq e 2 47r dft (du;/7rc<;)[p2 ' e /P2 ' (q/w) - Pi ' e/pi (q/co)]' (24-1) where o" is the scattering cross section (neglecting emission). Now consider an energetic Compton scattering in which a third, weak photon is emitted. The three diagrams are shown in Fig. 24-1. (weak) (weak) (weak) FIG. 24-1 Show that the cross section for this effect is given by Eq. (24-1), with the Klein-Nishina formula replacing cr . (Remember to assume q small.) potential region FIG. 24-2 FIG. 24-3 Interaction of Several Electrons Even though the Dirac equation describes the motion of one particle only, we can obtain the amplitude for the interaction of two or more particles from the principles of quantum electrodynamics (so long as nuclear forces are not involved). First consider two electrons moving through a region where a potential is present and assume that they do not interact with one another (see Fig. 24-2). The amplitude for electron a moving from 1— ~3, while electron b moves from 2 — * 4 is given the symbol K(3,4;l,2). If it is assumed that no interaction between electrons takes place, then K can be written as the product of kernels K + (a) (3,l) K + (b) (4,2), where the superscript means that K + (a) operates only on those variables describing particle a, and similarly for K + (b) . A second type of interaction gives a result indistinguishable from the first by any measurement in accordance with the Pauli principle. This dif- fers from the first case by the interchange of particles between positions 3 and 4 (see Fig. 24-3). Now the Pauli principle says that the wave function of a system composed of several electrons is such that the interchange of space variables for two particles results in a change of sign for the wave function. Thus the amplitude (including both possibilities) is K = K + (a) (3,l) K + (b) (4,2) - K + (a) (4,l) K + (b) (3,2). A similar situation arises in the following occurrence. Initially, one elec- tron moves into a region where a potential is present. The potential creates a pair. Finally one positron and two electrons emerge from the region. There are two possibilities for this occurrence, as shown in Fig. 24-4. Again, the total amplitude for the occurrence is the difference between the amplitudes for the two possibilities. 118 INTERACTION OF SEVERAL ELECTRONS 119 potential region FIG. 24-4 The probability of this occurrence, or the previous, or any other similar occurrence is given by the absolute square of the amplitude times the num- ber P v . The P v is actually the probability that a vacuum remains a vac- uum; because of the possibility of pair production, it is not unity. The P v can be computed by making a table of the probabilities of starting with noth- ing and ending with various numbers of pairs, as is shown in Table 24-1. TABLE 24-1 Final number of pairs Probability Pvl 2 1 P V |K + (2,1)| 2 2 P v |K + (3,1) K + (4,2) -K + (4,l) K + (3,2)| 2 3 etc. etc. The sum of all these probabilities must equal unity, and P v is determined from this equation. The magnitude of P v depends on the potential present. So the "probabilities" taken as merely the squares of amplitudes (that is, omitting the P v factor) are actually relative probabilities for various oc- currences in a given potential. Use of 6 + (s 2 ). For the present, the existence of more than one possibility for an occurrence (the Pauli principle) will be neglected. The total ampli- tude can always be derived from one by interchanging the proper space var- iables, making the corresponding changes in sign, and summing all the am- plitudes so obtained. 120 QUANTUM ELECTRODYNAMICS The nonrelativistic Born approximation to the amplitude for an interac- tion is K(3,4; 1,2) = K (0) + K (1) where, from earlier lectures, K (0) = K (a) (3,l)K (b) (4,2) and K (1) = -i/K (0) (3,4;5,6)V(5,6)K (0) (5,6; 1,2) d 3 X 5 d 3 X 6 dt 5 Note that t 5 = t 6 since a nonrelativistic interaction affects both particles simultaneously. The potential for the interaction is the Coulomb potential V(5,6) =e 2 /r 5)6 Separate variables may be used for t 5 and t 6 , if the function 6(t 5 — t 6 ) is included as a factor. Then K (1) --i//Ko(3,5)K (4,6)(e 2 /r 5 ,6)5(t5-t 6 )K (5,l)K (6,2) x dT 5 dT 6 where the differential dr includes both space and time variables. It is con- ceivable that the relativistic kernel could be obtained by substituting K + for K , and introducing the idea of a retarded potential by replacing 6(t 5 - t 6 ) by 6(t 5 - t 6 -r 56 ). However this 5 function is not quite right. Its Fourier transform contains both positive and negative frequencies, whereas a photon has only positive energy. Thus 6(X) = f 00 ^ exp(-iwX) dcj/27r To correct this, define the function 6 + (X) = /f exp(-icoX) dw/ir which contains only positive energy. The value, of the function is determined by the integral. Thus, 6 + (X) = lim (l/7ri)(X - ie) e-0 = 6(X) + (l/7ri)(principal value 1/X) Abbreviating t 5 - t 6 = t and r 5 6 = r, and taking account of the fact that both t 5 =£ t 6 and t 5 a: t 6 are possible, the retarded potential is INTERACTION OF SEVERAL ELECTRONS 121 V(5,6) = (e 2 /2r)[6 + (t-r) + 6 + (-t - r)] Exercises: (1) Show that (l/2r)[6 + (t - r) + 6 + (-t - r)] = 6 + (t 2 - r 2 ) Defining t 2 - r 2 as s 5 6 2 , a relativistic invariant, the potential is e 2 <5 + (s 5)6 2 ). Another term which must be included is the magnetic in- teraction, proportional to — V a • V b . In the notation used for the Dirac equation, this product is — a a • a h . It will be found convenient to ex- press this in the equivalent form -(/3oO a ■ (j3a) b , and in this notation the retarded Coulomb potential is proportional to /3 a j3 b . These (3 's come from the use of the relativistic kernel. Thus the complete po- tential for the interaction becomes e 2 6 + (s 5>6 2 )[/3 a - i S b -(/3a) a - (jS«) b ] = e 2 6(s 5>6 2 ) y/ a) r M (b) and then the first-order kernel is K< 1) (3,4;1,2) = -ie 2 //K + (a) (3,5) K + (b) (4,6)7 M (a) y/i (b) x 6 + (s 5>6 2 ) K + < a >(5,l) K + (b) (6,2) dr 5 dr 6 = -ie 2 //[K + (3,5)y M K + (5,l)] a 6 + (s 5>6 2 ) x [K + (4,6)y u K + (6,2)] b dr 5 dr 6 (24-2) Here the superscript on y M indicates on which set of variables the matrix operates, just as for the superscripts on K + . The occurrence represented by this kernel can be diagrammed as in Fig. 24-5. This represents the exchange of a virtual photon be- FIG. 24-5 122 QUANTUM ELECTRODYNAMICS tween the electrons. The virtual photon can be polarized in any one of the four directions, t, x, y, z. Summation over these four possibilities is indicated by the repeated index of 7^7^. The integral expression for the kernel, Eq. (24-2), implies that the amplitude for a photon to go from 5 — 6 (or from 6 — * 5 depending on timing) is 6 + (s 56 2 ). Equation (24-2) can be taken as another statement of the fundamental laws of quantum electrodynamics. (2) Show that 6 + (s 2 ) =-47rJ[exp(-ik-X)] d 4 k/(k 2 + ie)(27r) 4 Thus, in momentum space, 6 + (s 2 )— -4?r/k 2 Twenty -fifth Lecture DERIVATION OF THE "RULES" OF QUANTUM ELECTRODYNAMICS From the results of the last lecture, it is evident that the laws of electro- dynamics could be stated as follows: (1) The amplitude to emit (or absorb) a photon is ey^ , and (2) the amplitude for a photon to go from 1 to 2 is ^+( s i,2 2 )> where 6 ( S 2>=-47r fc - ik '< x 2- x i) d4k / 25 ,v MHl>- 47T j e ^ - - —r (25 1) = -4?r/(k 2 + ie) in momentum representation. It is interesting to note that S + (s 1>2 2 ) is the same as I + (s lj2 2 ), the quantity appearing in the derivation of the propagation kernel of a free particle, with m, the particle mass, set equal to zero. A more direct connection with the Maxwell equations can be seen by writing the wave equation, □ 2 A fi = 47r J^ in momentum representation, -k 2 a M = 4ttj m or sl u = -(47r/k 2 ) j^ (25-2) We now consider the connection with the "rules" of quantum electrody- namics given in the second lecture. The amplitude for a to emit a photon which b absorbs will now be calculated according to those rules (see Fig. 25-1). The amplitude that electron a goes from 1 to 5, emits a photon of polarization i and direction K, then goes from 5 to 3 is given by [K + (3,5) i V(47re 2 /2K) exp(-iK'r 5 ) exp (iKt 5 )K + (5,l)] a INTERACTION OF SEVERAL ELECTRONS 123 whereas the amplitude that b goes from 2 to 6, absorbs a photon of polari- zation 4 and direction K at 6, then goes from 6 to 4 is given by [K + (4,6) * V(47reV2K) exp (iK • r 6 ) exp (-iKt 6 ) K + (6,2)] b The amplitude that both these processes occur, which is equivalent to b ab- sorbing a's photon if t 6 > t 5 is just the product of the individual amplitudes. If a absorbs b's photon, the signs of all the exponentials in the preceding amplitudes are changed and t 6 must be less than t 5 . To obtain the amplitude that any photon is exchanged between a and b, it is necessary to integrate over photon direction, sum over possible photon polarizations, and integrate over t 5 and t 6 , subject to the aforementioned restrictions. In summing over polarizations, $ will be replaced by y„ and a summation over pt will be taken. This amounts to summing over four di- rections of polarization, something that will be explained later. Thus f Amp. fori \ photon \ = 47re 2 £ J exp [-iK ■ (r 5 - r 6 )l exp [iK(t 5 - 1 6 )] [a — b J " x [K + (3,5)y M K + (5,l)] a [K + (4,6)^K + (6,2)] b x (l/2K)[d 3 K/(27r) 3 ] dt 5 dt ( t 6 >t 5 47re 2 2 /exp [iK ■ (p 5 - r 6 )] exp [-iK(t 5 - t 6 )j u x [K + (3,5)y^K + (5,l)]JK + (4,6)y p K + (6,2)] b x (l/2K)[d 3 K/(27r) 3 ] dt 5 dt 6 t 6 < t 5 (25-3) 124 QUANTUM ELECTRODYNAMICS Comparing this with the result of the last lecture, it must be that <5 + (s 5 , 6 2 ) = 4tt J' exp [-iK ■ (r 5 - r 6 )] exp [iK(t 5 - 1 6 ) (1/2K) x [d 3 K/(27T) 3 ] t 6 >t 5 = 4tt j exp [iK • (r 5 - r 6 )] exp [-iK(t 5 - 1 6 )] (1/2K) x [d 3 K/(27i) 3 ] t 6 <t 5 This can be written in a form which makes the space -time symmetry evi- dent by using the Fourier transform exp(-iK|t|) = J2o [2iK/(w 2 -K 2 + ie)] exp (-iwt) dw/27r so that the foregoing equation becomes 6+(s56 2 ) = ^ 47r / expt-ik-(x 5 -x 6 )] d^k + 5 ' 6 ' J k 4 2 -K-K+ie (2ti) 4 (25-4) and comparing this with the result of the last problem of Lecture 24 estab- lishes that the rules given in Lecture 2 are consistent with relativistic elec trodynamics developed in the last lecture. ELECTRON-ELECTRON SCATTERING The theory will now be used to obtain the electron-electron scattering cross section. The diagrams for the two indistinguishable processes are shown in Fig. 25-2. & FIG. 25-2 INTERACTION OF SEVERAL ELECTRONS 125 The amplitude expressed in momentum representation is obtained as follows: Write Eq. (25-3) [with the aid of Eq. (25-4)] as e 2 E/[ K + ( 3 ' 5 )^K + (54)] a ^[K + (4,6)y M K + (6,2)] b ^4 x dT 5 dT 6 Since electron state 1 is a plane wave of momentum ^i and electron state 3 is a plane wave of momentum $ 3 , it is clear that in momentum representa- tion the spinor part of the first bracket will become (i^y^Ut) and the spinor part of the second bracket will become (u 4 y^u 2 ). Integration over T5 and tq produces the conservation laws given at the bottom of the diagrams. Drop- ping the integration over q puts the photon propagation in momentum repre- sentation directly. Thus the matrix element can be written M = + i47re 2 (u 4 y ti u 2 )(u 3 y a u 1 ) (u 4 y u u 1 )(u 3 y |tx u2) (fa " fa) 2 (fa " fa) 2 The first term comes from diagram R, the second from diagram S, and the summation over /u is implied. In the center-of-mass system, the probability of transition per second is Trans, prob./sec = crvt = 7^74 l M l /0 _ 3 OT7 ,_2 (2E)* '"""' (2;r) 3 2Ep' (see Density of Final States, Lecture 19). The method of Lecture 23 can be used to average over initial spin states and sum over final spin states. For example, the sums over spin states that result from R by R matrices and R by S plus R by S matrices are RR [ Sp[fa + m)y^ 2 + m)yJSp [fa + my^fa + m)y y [(fa -fa) 2 ] 2 Sp [fa + m)y y fa + m)y„ fa + m)y v fa + m)y fa - fa) 2 fa - far By judicious use of the spur relations given in Lecture 23 the following dif- ferential cross section is obtained (alternatively, Table 13-1 could be used to calculate M directly): 2e 4 p dft 4x 2 + 8x cos + 2(1 - cos 2 9) + 4 cos 9 (1 - cos 9) *fl) -4cosl (1 + COS0) 2 (l-cos0)(l + cos#)_ 4x 2 - 8x cos 0+2(1- cos 2 9) -4 cos 9 4(1 + x)(x - 3) 126 QUANTUM ELECTRODYNAMICS E,p FIG. 25-3 where x = E 2 /p 2 - This is called Moller scattering (see Fig. 25-3). Problems: (1) Calculate positron-electron scattering by the pre- ceding method. (2) Find the cross section for a \x meson to produce a knock-on electron. Assume that the fi meson satisfies the Dirac equation with S = 1/2 and no anomalous moment. Remember that the particles are distinguishable and hence there is no interchange of particles. (3) Calculate the expected electron-proton scattering cross section assuming the proton has no structure but does have an anomalous moment. The Dirac equation for a proton is (see page 54) (if+ M-eJ)L -{n/4M)y u y v F^)^= Of Thus the perturbing potential can be taken as (see page 54) eJjL + (e M /4M)y M y y (V„ A v - V V A^) and the coupling with a photon is e^ + (e M /4M)(^ - e^) or ey^ + (e M /4M)(^y M -y^i) The Sum over Four Polarizations. In classical electrodynamics, longitu- dinal waves can always be eliminated in favor of transverse waves and an instantaneous Coulomb interaction. This is the approach used by Fermi (see Lecture 1), and it will now be demonstrated that the sum over four polariza- tions is also equivalent to transverse waves but plus an instantaneous Cou- lomb interaction. If instead of choosing space directions x, y, z, one direc- tion parallel to Q (photon momentum) and two directions transverse to Q are taken, the matrix element can be written t For the proton y, = 1.7896. INTERACTION OF SEVERAL ELECTRONS 127 M/-i47re 2 = (u 4 y t u 2 )(l/q 2 )(u 3 y 1: u 1 ) - ( u 4 y Q U2)(l/q 2 )(u 3 'y Q u 1 ) " £ (u 4 y tr U2)(l/q 2 )(u 3 y tr u 1 ) 2 tr. direc. where yg is the y matrix for the Q directions and y tr represents the y ma- trix in either of the transverse directions. The matrix element of ^ = q 4 y t - Qyp is zero in general (from the argument for gauge invariance). f Thus yg can be replaced by (q 4 /Q)y t with the result ^2 = (u A y t u 2 )^ (l-^hu 3 y t Ui) - £ KTtrUa) ~2 (u a r tt Ui) = -(U 4 y t u 2 )^2 (Wt u l) ~Tj (^47tr u 2)l2 (u 3 y t rUi) ^ 1,2 * Now 1/Q 2 represents a Coulomb field in momentum space and y t is the fourth component of the current density or charge, so that the first term represents a Coulomb interaction while the second term contains the inter- action through transverse waves. tin our special case, it is easy to see directly, for example, (U 4 ^U 2 ) = (U 4 $ 2 -^4)U 2 ) = (^4^2 U 2) " (U 4 ^4 U 2) = m(u 4 u 2 ) - m(u 4 u 2 ) = Discussion and Interpretation of Various "Correction" Terms Twenty -sixth Lecture In many processes the behavior of electrons in the quantum -electrody- namic theory turns out to be the same as predicted by simpler theories save for small "correction" terms. It is the purpose of the present lecture to point out and discuss a few such cases. ELECTRON-ELECTRON INTERACTION The simplest diagrams for the interaction are shown in Fig. 26-1. The amplitude for the process has been found to be proportional, in momentum 128 "CORRECTION" TERMS 129 representation, to (u 3 r fi u 1 )(u 4 r 1L[ u 2 )/q 2 where q = (Q, q 4 ) and Q is the momentum exchanged by the two electrons. Also, since sk = F^i ~ $i it follows that (l^Ui) = (U 8 (A - F^Ut) = From this identity it was deduced in the last lecture that the amplitude for the process as just given is equivalent to [-(u 3 y t ui)(u 4 y t u 2 )/Q 2 ]-^ (u 3 ytrUi)(u 4 y tr U2)/q 2 1,2 By taking the Fourier transform of the first term, it can be seen that it is the momentum representation of a pure, instantaneous Coulomb potential. The second term then constitutes a correction to the simple Coulomb inter- action. In it y tr denotes the y's for two directions transverse to the direc- tion of Q. For slow electrons, the correction to the Coulomb potential may be sim- plified and interpreted in a simple manner. Note that in this case Q =Pl -Ps and q 4 = Ei - E 2 « [m+ (Pi 2 /2m)] - [m + (p 3 2 /2m)] = ( Pl 2 -p 3 2 )/2m = [(Pi + p 3 )/2m](pi - p 3 )~ v(pi - p 3 ) so that q 4 2 ~ v 2 Q 2 and q 2 in the denominator can be replaced by -Q 2 with small error. (In the C.G. system, q 4 = exactly.) The correction term be- comes + Tj (^3rtrU 1 )(u 4 y tr u 2 )/Q 2 1,2 but (WtrUi) = u 3 *a tr u 1 It is recalled that u = I a J, where u a is the large part and u b the small part and that in the nonrelativistic approximation u b « (l/2m)(a • n)u a 130 QUANTUM ELECTRODYNAMICS Also, since -C':) it follows that (taken between positive energy states) = l/2m[u 3 * (7(a-n 1 )+(<7-n 3 )(7u la ] tr In free space II = p, so the x component, for example, of the foregoing ma- trix is M<*xPlx + °y Ply + u z Plz) + (°"xP3x + °y P3y + °z P3z )°x " (Pi + P3)x + i[cr z (Pl-Ps)y " Oy(Pi-Ps) x ] where the commutation relations for the cr's have been used. From this it is easily seen that the amplitude for the correction to the Coulomb potential may be written altogether in the form S<?H X J u l 4a P1+P3 _ t O-X (Pi -Ps ) 2m 2m P4 + P2 _ . O-X (P2 ~ P4) 2m 2m "la} U 2 ; tr The first terms in each of the brackets represent currents due to motion of the electron transverse to Q and the second terms represent the transverse components of the magnetic dipole of each. So altogether it appears that the correction arises from current-current, current-dipole, and dipole -dipole interactions between the electrons. These interactions are expected even on the basis of classical theory and were described by Breit before quantum electrodynamics, hence are referred to as the Breit interaction. Consider the dipole-dipole term arising in the correction factor. Since Q = Pi " P3 = P2 - P4 it; is E (<7l*Q)tr (CT 2 XQ)tr/Q 2 1,2 But since (jXQ is zero when a and Q have the same direction, the sum could as well be over all three directions and then it is equivalent to a dot product. That is, this term of the correction is fa XQ)- (cr 2 xQ)/Q^ CORRECTION" TERMS 131 By taking the Fourier transform f this will be seen to be the momentum representation of the interaction between two dipoles as was stated. Note that the approximation q 4 ~ (v/c)Q used above applies only between positive energy states. For, if one of the states represents a positron, then q 4 = Ej - E 2 * = 2m However, 2m is very large, so the correction is still small. It is necessary to redo the analysis nevertheless. ELECTRON-POSITRON INTERACTION It would appear that, since the electron and positron are distinguishable, the Pauli principle would not require the interchange diagram, leaving as the only one Fig. 26-2. FIG. 26-2 But it is still possible by the same phenomenological reasoning to con- ceive of the diagram in Fig. 26-3, which would represent virtual annihilation of the electron and positron with the photon later creating a new pair. It turns out that it is necessary to regard an electron-positron pair as exist- ing part of the time in the form of a virtual photon in order to obtain agree- ment with experiment. t Notice that (a^ x Q) • (<y 2 X Q) exp (-iQ • x), which will appear in trans- form integral, is the same as -(^i x V ) • (<r 2 * V) exp (-iQ ■ x), where V is the grad operator. This device enables an integration by parts, which greatly simplifies the process and the result. Thus, since the transform of 1/Q 2 is 1/r, the coupling is -(oi x V) • (<t 2 x V)(l/r), which is the classical energy for interacting magnetic dipoles. 132 QUANTUM ELECTRODYNAMICS FIG. 26-3 From the point of view that positrons are electrons moving backward in time, Fig. 26-3 differs from Fig. 26-2 only in the interchange of the "final" states ^ 3 , ^ 4 . The Pauli principle extended to this case continues to oper- ate; the amplitudes of the two diagrams must be subtracted, since they dif- fer only in which outgoing (in the sense of the arrows) particle is which. POSITRONIUM An electron and positron can exist for a short time in a hydrogenlike bound state known as the atom positronium. The ground state of positronium is an S state and may be singlet or triplet, depending on the spin arrange- ment. As has been indicated in assigned problems, the *S state can anni- hilate only in two photons, whereas the 3 S state decays only by three -photon annihilation. The mean life for two-photon annihilation is 1/8 x 10 9 sec and for three photons it is 1/7 x 10 6 sec. Problem: Check the mean life 1/8 x 10 9 sec for two-photon anni- hilation using the cross section already computed and using hydrogen wave functions with the reduced mass for positronium. Figure 26-2 contributes the Coulomb potential holding the positronium together. The correction term (Breit's interaction) arising from this same diagram contributes a dipole-dipole or spin-spin interaction that is different in the 3 S and *S states (the current-current and spin-current interactions are the same for both states). Thus this amounts to a fine-structure sepa- ration of the 3 S and S states which can be shown to be 4.8 x 10 ev. In view of the fact that a photon has spin 1, and the *S state of positro- nium spin 0, conservation of angular momentum prohibits the process in Fig. 26-3 from occurring in the *S state. It does occur in the 3 S state, however. The term arising from this diagram is small and, therefore, constitutes an- other fine-structure splitting of the 3 S and S levels. It can be shown to CORRECTION" TERMS 133 amount to 3.7 x 10 ev in the same direction as the spin-spin splitting. It is referred to as splitting due to the 'mew annihilation force." In order to calculate the term arising from Fig. 26-3, one needs to com- pute -(u 4 y JLl u 1 )(u3r |i U2)/q 2 In this case q w 4m (Q = in the C.G. system), and all matrix elements are 1 or (regarding particles as essentially at rest in the positronium), so the result is just a number. This means that taking the Fourier transform one gets a 6 function of the relative coordinate of the electron and positron for the interaction in real space. For this reason it is sometimes referred to as the "short-range" interaction of the electron and positron. The combined fine -structure splitting due to the effects already outlined turns out to be represented by (1/2) a 2 Rydberg (7/3) where a is the fine -structure constant. This amounts to 2.044 x 10 5 Mc, using frequency as a measure of energy. There is still another correction, however, not yet mentioned, arising from diagrams, such as Fig. 26-4, where the electron or positron may emit FIG. 26-4 and reabsorb its own photon. Taking this into account, the fine-structure splitting in positronium is given by f (1/2) a 2 Rydberg {(7/3) - [(32/9)+ 2 In 2] (a /it)} tPhys. Rev., 87, 848 (1952). 134 QUANTUM ELECTRODYNAMICS having a value of 2.0337 x 10 5 Mc. The experimental value for the positro- nium fine structure is 2.035 ± 0.003 Mc, so it is seen that this last correction, though of order a smaller than the main terms, is necessary to obtain agreement with experiment. It is referred to both in positronium and in hy- drogen as the Lamb-shift correction because of its experimental observa- tion by Lamb as the source of the small splitting between the 2 Sj/ 2 and 2 Pi/2 levels in hydrogen. In general, it comes under the heading of self-action of the electron, to be treated in more detail later. TWO-PHOTON EXCHANGE BETWEEN ELECTRONS AND/ OR POSITRONS It is easy to imagine that processes, indicated by the diagrams in Fig. 26-5, may occur where two photons instead of one are exchanged. Although FIG. 26-5 it has not been necessary to consider such high-order processes to secure agreement with experiment, it may become necessary as experimental re- sults improve. The amplitudes for the processes may be written down easily but their calculation is difficult. The amplitude for case II in space -time representation is, for example, -e 4 f JJS [K + (3J)y v K + (7,5)y M K + (5,l)l [K + (4,8)r fi K + (8,6)y, x K + (6,2)] S + (s 2 7j6 )<5 + (s 2 5, 8 ) dT 5 dT 6 dT 7 dT 8 or in momentum representation it is -(4^)V/(u 3 y, ^_ K 1 i _ m r M u 1 )(u 4 y ^ 2 _ K 1 2 _ m y t/ u 2 ) d*fe ki 2 k 2 2 (27r) 4 CORRECTION" TERMS 135 FIG. 26-6 where ii ~ ti + Ki = rf 4 or % = A + Ki ~ & (see Fig. 26-6). Thus it is possible to determine #i and #2 in terms of each other but not independently; that is, the momentum may be shared in any ratio between the two photons. It is for this reason that the integral over #i arises in the expression for the amplitude. Twenty -seventh Lecture SELF-ENERGY OF THE ELECTRON t In Lecture 26 the following idea was introduced: An electron may emit and then absorb the same photon, as in Fig. 27-1. Then the propagation ker- nel for a free electron moving from point 1 to point 2 should include terms representing this possibility. Including only a first-order term (only one photon is emitted and absorbed), the resulting kernel is K(2,l) = K + (2,l) - ie 2 /J K + (2,4)y„K + (4,3)y„K + (3,l)6 + (s 4 , 3 2 ) x dT 4 dT 3 (27-1) The correction term in this equation is written down by an inspection of the diagram, following the usual procedure for scattering processes. In the present case, the initial and final momenta are identical. Therefore the |R. P. Feynman, Phys. Rev., 76, 769 (1949); included in this volume. 136 QUANTUM ELECTRODYNAMICS nondiagonal elements in the perturbation matrix will all be zero. A diagonal element is one in which the resulting wave functions of a particle remain in the same eigenstate. For time -independent perturbations, it was shown in the development of perturbation theory that the only effect on such wave func- tions is a change in phase, proportional to the time interval T over which the perturbation is applied. The resulting wave function is exp (-iE n T) exp[-i(AE)T] (27-2) Since the perturbation effect (AE)T is small, the second exponential can be expanded as 1 - i(AE)T + ••• and higher-order terms neglected. It is the second term of this expansion which is represented by the integral on the right side of Eq. (27-1). The representation is not yet an equality, since certain normalizing factors are different in the two expressions. K + (3,l) FIG. 27-1 To obtain the correct equation proceed as follows: First, it is clear that the probability of the occurrence depends only on the interval in space and time between points 3 and 4, and not at all on the absolute values of the space and time variables. So suppose a change of variable is made so that dT4 represents the element of interval (in space and time) between 3 and 4. Then write the integral in Eq. (27-1) //f(4)y M K + (4,3)y M 6 + (s 4)3 2 )f(3)dT 4 d 3 x 3 dt 3 (27-3) where it is clear that the operators K + and 6+ depend only on the interval 3-4. Second, expression (27-2) contains the time -dependent part of the wave function, exp (-iE n t), because it was assumed that the wave functions used did not contain time factors. In Eq. (27-3), f(3), f(4) do already include the time -dependent part, so it should be omitted in Eq. (27-2). "CORRECTION" TERMS 137 Third, the normalization of wave functions is different for the two ap- proaches. For the development that led to Eq. (27-2), the normalization /*** dv = 1 was used. For the present development the normalization is j u* u dv = (2E/cm 3 ) • V (27-4) Thus, to establish an equality, expression (27-3) must be divided by the nor- malizing integral of Eq. (27-4). The resulting expression is -ie 2 //f (4)y u K + (4,3)y u 6 + (s 4 , 3 2 )f(3) dr 4 d 3 x 3 dt 3 2E-V The integral over d 3 x 3 gives a V which cancels with the denominator, and the integral over dt 3 gives a T which cancels with the left-hand side, so finally 2E AE =+e 2 / uy M K + (4,3)y M 6 + (s 4)3 2 )u dr 4 (27-5) Note that the integral is relativistically invariant. Further, since p is the same before and after the perturbation and E 2 - m 2 + p 2 , the change in E can be taken as a change in the mass of the electron, from 2E AE = 2m Am Using this expression, and transforming to momentum space, 47re 2 r~( l \ d 4 k 1 /or7 ox Am = - — r J u y u -j — -, y u u -—7 72 (27-6) 2mi J \ V p-K-m Vj (2iry k^ The integrand may be rewritten from 1 y u (^-^ + m)y fJ _2_ m + 2 }L 'Vtf-K-m'l 1 ^-2p-k + k'-m' i k'-2p-k using ^u = mu and the relations of Lecture 10. Then Eq. (27-6) becomes 4?re 2 C 2m + 2]£ d 4 k 1 /97 RM Am = I —9 — t —9 (27-6') i J k 2 -2p-k (2tt) 4 k 2 V This integral is divergent, and this fact presented a major obstacle to quantum electrodynamics for 20 years. Its solution requires a change in the fundamental laws. Thus suppose that the propagation kernel for a photon is 138 QUANTUM ELECTRODYNAMICS (l/k 2 )c(k 2 ) instead of just (1/k 2 ), where c(k 2 ) is so chosen that c(0) = 1 and In space representation the modification takes the form c(k 2 )-~0 as k 2_~« <5 + (si2 2 )-^f + (s 1)2 2 )= J(l/k 2 )c(k 2 )exp(-ik-x)d 4 k/(27r) 4 (27-7) The new function f + differs significantly from 6 + only for small inter- vals. This is clear from the fact that if the high-frequency components are removed from the Fourier expansion of a function, only the short-range de- tails are modified. In the present case the size of the interval over which the function is modified can be described roughly as follows: Consider a large number, X , and suppose that so long as k « A.' c(k 2 ) 1. Then (from the exponential term) differences will occur when the interval s^» 1/X . Call FIG. 27-2 this value a , and the general behavior of f + is shown by Fig. 27-2. Thus a' is sort of a ''mean width" of f , . If a 2 « 1, as assumed, then when t 2 -r 2 a 2 /2r (27-8) which is the size of the interval. The significance of the form of f + (s ) can be understood from the following. The original function, <5 + (s 2 ) differs from zero only when s = t — r = 0. That is to say, an electromagnetic signal can reach a point at distance r only at a time t such that t 2 - r 2 = or t = r (i.e., the speed of light is 1). This is no longer true for f + (s 2 ). The depar- ture is obtained by a measure of t - r. But, by Eq. (27-8), for all values of r »a this measure is negligible. Thus, depending on A 2 , the laws will be found unaffected over any practical distance. "CORRECTION" TERMS 139 Choosing A 2 » m 2 , a practical (and general) representation of c(k 2 ) is c(k 2 ) = Jg(X) dX(-X 2 )(k 2 - A 2 )" 1 and the simple form is suggested, c(k 2 ) = -X 2 /(k 2 - A 2 ) From this, obtain the propagation kernel as l/k 2 (-A 2 )(k 2 -A 2 ) _1 = 1/k 2 - l/(k 2 -A 2 ) The second term is that for the propagation of a photon of mass X; how- ever, the minus sign in front of the term has not been explained so far from this point of view. A convenient representation for this kernel is the integral r x 2 -J o dL/(k 2 - L) 2 (27-9) Introducing this kernel into Eq. (27-6' ) in place of 1/k 2 gives f 2m + 2k djk/ -X 2 \ J k 2 - 2p • k k 2 U 2 - X 2 ) (27 " 10) which can be written as the sum of two integrals, which differ only by having m or ^ in the numerator, that is, m or k (since # = k y ). METHOD OF INTEGRATION OF INTEGRALS APPEARING IN QUANTUM ELECTRODYNAMICS We shall need to do many integrals of a form similar to the preceding one. A method has been worked out to do these fairly efficiently. We now stop to describe this method of integration. Everything will be based on the following two integrals:! roo (l;k )d 4 k _ J /0 \ n 2 . — 3 = (327T 2 iL)- 1 (l;0) (27-11) J -°° (27r) 4 (k^ + ie - L) d v ' v ' v ; J [ax + b(l -x)]" 2 dx = 1/ab (27-12) In Eq. (27-11), to write a little more compactly, we use the notation (l;k ) to mean that either 1 or k Q is in the numerator, in which case, on the right- hand side the (1;0) is 1 or 0, respectively. To prove the first of these, note •j-R. P. Feynman, Phys. Rev., 76, 769 (1949); included in this volume. Note that in the article d 4 k is equivalent to 47r 2 [d 4 k/(27r) 4 ] in our notation. 140 QUANTUM ELECTRODYNAMICS that, if k is in the numerator, the integrand is an odd function. Thus the integral is zero. With 1 in the numerator, contour integration is employed, Write the integral Then for e « L + k 2 , there are poles at cu= ±[(L + k 2 ) l/2 - ie], and contour integration of lo gives J°° [(J + ie - (L + k 2 ) J" 1 dw = 2?ri[ -2(L + k 2 ) _l/2 ] with the contour in the upper half -plane. Two differentiations with respect to L give f_ °°Ju> 2 + ie - (L + k 2 )] " 3 da; = (67r/16i)(L + k 2 )" 5/2 Then the remaining integral is //~/(L + k 2 )" 5/2 d 3 k = 4?r /°(L+k 2 r 5/2 k 2 dk = 4?r [k 3 /3L(L+ k 2 ) 3/2 ] | °° = 4?r/3L which proves Eq. (27-11). If k-p is substituted for the variable of integra- tion in Eq. (27-11), the result is r oo (l;k )d 4 k L (27r) V_ 2p . k _ A) 3 = I 32 - 2 HP 2 * A)l "' (l ; Po) (27-13) By differentiating both sides of Eq. (27-13) with respect to A or with respect to Pj , there follows directly roo (l;k ;k kj)d 4 k = _ [l;p ;p pj - (l/2)6 oJ (p 2 + A)] ^-« (2tt) 4 (k 2 - 2p • k - A) 4 96tt 2 i(p 2 + A) 2 Further differentiations give directly successive integrals including more k factors in t nominator. factors in the numerator and higher powers of (k 2 - 2p • k - A) in the de Twenty -eighth Lecture SELF-ENERGY INTEGRAL WITH AN EXTERNAL POTENTIAL Last time it was found that the self-energy of the electron is equivalent to a change in mass CORRECTION" TERMS 141 Am "2ml j k 2 - 2p • k U 2 - A 2 J k 2 (27T) 4 (28 ~ lj and that this could also be expressed in terms of integrals, /A 2 , (l;k ) d 4 k 1 = "Jo dL J (k 2 -2p-k)(k 2 -L) 2 ^ (28 " 2) It was also found that r (ljk ) <^ 9 12 1 f (k 2 -2p-k-A) 3 = (32?r i} (P + A) (28 " 3) Using the definite integral JL _ f 1 _ 2(i-x)dx ab 2 " j o [ax + b(l-x)] 3 (28 4) the denominator of the integrand of Eq. (28-2) may be expressed as = 1 2(1 -x) dx (k 2 - 2p • k)(k 2 - L) 2 J o [k 2 - 2xp • k - L(l - x)] 3 so that Eq. (28-2) becomes A 2 , ,1 d 4 k(l;k G )2(l - x) dx I=-J dL J i [k 2 - 2xp • k - L(l - x)] 3 (2^ (28_5) The integral over k can be done by using Eq. (28-3) with the substitutions xp for p and L(l — x) for A, giving I = -X dL I X 2 ,1 (l;pJ2(l -x)dx 2 2 TTT73272TT7. ^T P =m 'o ^o [327rM][x^p^+ L(l -x)] The integral over L is elementary and gives 1 = -2(327r 2 i) _1 J dx(l;xp ) ln[(l-x)A 2 +m 2 x 2 /m 2 x 2 ] When A 2 »m 2 , it is legitimate to neglect m 2 x 2 in the numerator [it is true that when x~ 1, (1 - x)X 2 is not much larger than m 2 x 2 , but the interval over which this is true is so small, for A 2 »m 2 , that the error is small], so that, when the x integration is performed,! r 1 r 1 t 1 In [x" 2 (l -x)] dx= 1 J x ln[x" 2 (l - x)] dx = -1/4 142 QUANTUM ELECTRODYNAMICS I w -(327r 2 i)- 1 {2[ln(\ 2 /m 2 )+ 2]; p [ln(\ 2 /m 2 ) - 1/2]} X »m 2 The change in mass is [from Eq. (28-1)] Am= (47r 2 /2mi)(-327r 2 i)- 1 (u{2m[2 ln(\ 2 /m 2 ) + 2] + 2^[ln(A 2 /m 2 ) - (l/2)]}u) Since ^u = mu and (uu) = 2m, this can be simplified to Am/m = (e 2 /27r) [3 In (\/m) + (3/4)] (28-6) Now (e 2 /27r) is about 10~ 3 , so that even if A. is many times m, the fraction change in mass will not be large. The interpretation of this result is as fol- lows. There is a shift in mass which depends on k and hence cannot be de- termined theoretically. One can imagine an experimental mass and a theo- retical mass which are related by m exp =m th +Am < 28 " 7 ) All our measurements are of m , that is, self-action is included, and m th the mass without self-action, cannot be determined. More accurately stated, 'A theory using m^ and e 2 /fic self-action {a theory using m exp , plus e 2 /hc self-action minus Am as computed for a free particle When the electron is free, the e 2 /fic self-action term exactly cancels the Am term and a theory using m exp is exactly correct. When the electron is not free, e 2 /Rc self-action is not quite equal to the Am term and there is a small correction to a theory using m ex p • This effect leads to the Lamb shift in the hydrogen atom, and, in order to calculate such effects, we shall now consider the effect of self-action on the scattering of an electron by an external potential. SCATTERING IN AN EXTERNAL POTENTIAL The diagram for scattering in an external potential is shown in Fig. 28-1, and the relationships for this process, excluding the possibility of self- action, are as follows: Potential: a'(q) = y t (47rZe/Q 2 )6(q4) for Coulomb potential Matrix element: M = -ie( u 2 ^ u i) Conservation relation: ^ 2 = K^i + & CORRECTION" TERMS 143 First-order self-action will produce the diagrams shown in Fig. 28-2. The amplitude for process is obtained in the usual manner. For example, dia- gram I gives Rationalizing the denominators and inserting the convergence factor, this becomes _ 47re 2 r (^27 u ^2 -K+m] £\$ x -K+m]y u ui) 1 i j """"' (k 2 - 2p 2 • k)(k 2 - 2 Pl • k) -y k 2 -X 2 x(2,r^ (28-8) This expression also happens to diverge for small photon momenta (k) (a result which has been called the "infrared catastrophe," but which has a 144 QUANTUM ELECTRODYNAMICS clear physical interpretation, discussed later). Temporarily the k 2 under d 4 k will be replaced by (k 2 - A. 2 m i n ), where A. 2 m i n « m 2 , to make the inte- gral convergent. This is equivalent to cutting off the integral somewhere near k = \ m i n , and the physical interpretation is left to Lectures 29 and 30. To facilitate the integration over k, the following identity is used: x 2 - h . (k 2 - L)- 2 dL mm k 2 "A 2 min " k 2 -A 2 X 2 - A 2 A min A -X 2 (k 2 " A 2 min )(k 2 - X 2 ) ~ 1 k 2 -X 2 k — X min 2 ss ™2 \s \ 2 since X » m » X m - m . This substitution produces integrals of the form r^ 2 f (l;k ;k k r )(27r)-«d«k j ^ 2 mi n J (k 2 - 2 Pl • k)(k 2 - 2p 2 • k)(k 2 -L) 2 To evaluate these integrals, we make use of the identity (ab)- 1 ^ f dy/[ay+b(l-y)] 2 so that dz. (k 2 - 2pj • k)(k 2 - 2p 2 • k) J o (k 2 - 2^ y • k) 2 where $ y = y^j + (1 - y)^ 2 • Performing integrations in the order, k, L, y, and using the appropriate integrals in Eq. (28-6) gives as the matrix to be taken between states u 2 and Uj M,-* r 2 fln-^--lV 1 --^-V^an L \ Amin A tan 20/ 27rL"\'"X m in tan 20 tan 20 r e x I a tan a da J 27T to «* - «) T^e + ri . (28-9) where r = ln( X/m) +9/4-2 In (m/A min ) and 4m 2 sin 2 = q . It is shown in Lecture 30 that diagrams II and III (Fig. 28-2) produce a contribution M 2 + M 3 = -(e /2ir)r£ , which just cancels a similar term in M 3 . When q i mated by When q is small, 6 » (q 2 ) 1/2 /2m, and the sum M^ + M 2 + M 3 can be approxi CORRECTION" TERMS 145 M e 47T 2M<*-««» + iS?-> \ ^min 8/ (28-10) The (gfeL - 4<fy can be written out (M - M = y^Jy (qj,a y - a^q y ) But q^ is the gradient operator so this can be written, in coordinate repre- sentation, yflv ^n A v - v v A u ) = + y^v Y \iv [see Eq. (7-1)] . Reference to page 54 shows that the effect of a particle's having an anomalous magnetic moment is to subtract a potential [xy^Jy F„ y from the ordinary potential i. = y^ A^ appearing in the Dirac equation. Since this is precisely what the first term of Eq. (28-10) does, one can say that this part of the self-action correction looks like a correction to the elec- tron's magnetic moment, so that Melee = (e/2m)[l+ (e 2 /27r)] Note that this result [and (28-9) and (28-10)] does not depend on the cutoff X, and hence X can now be taken to be infinity.! Twenty -ninth Lecture It has been shown that when a particle is scattered by a potential, the pri- mary effect is that of $., and that for diagram I (Fig. 28-2) a correction term arises which is FIG. 28-2 t R. P. Feynman, Phys. Rev., 76, 769 (1949); included in this volume, 146 QUANTUM ELECTRODYNAMICS sH'"e-')( i -i^i)*"»» x*. x i ol tan a tan 26 i + 87rm (<$ - ii) 28 sin 20 .2 27T It remains to show that the combined effect of diagrams II and III (Fig. 28-2), I II FIG. 28-2 when considered along with the effect of the mass correction, is another correction term, -{e 2 /2ii)v£ just canceling the last term in the preceding expression. It is recalled that the necessity for considering the effect of the mass correction together with the self-action represented in diagrams I, II, and III is that the theory being developed must contain the experimental mass rather than the "theoretical" mass. Suppose that in the Dirac equation (if - m th )* = e$* m t h , the theoretical mass, is replaced by m - Am, where m is the experi- mental mass: then (if - m)# = e(4(+Am)* The mass correction Am is just a number, so that in momentum represen- tation it is a 6 function of momentum. Hence from the form of the foregoing equation, it is seen to behave like a potential with zero momentum and in- volves no matrices. Diagrammatically its effect may be represented as in Fig. 29-1. The minus sign is used because the effect of the mass correction Am is to be subtracted from the results obtained from diagrams I, II, and III (Fig. 28-2) alone. For diagram II the amplitude would appear to be CORRECTION TERMS 147 FIG. 29-1 ~ 1 /47re 2 f 1 1 and for diagram II' (Fig. 29-1), d 4 k -A (2tt) 4 k 2 - X 2 ■u 2 ;([l/(|i r m)](Am)u But the part of the amplitude for diagram II (Fig 28-2) contained in the pa- rentheses is just Amu 1} so that II and II' seem to cancel. A similar result applied for diagrams III and III'. This is an error, however, arising from the fact that both of these amplitudes are infinite, owing to the factor j^-m in the denominator. Hence their difference is indeterminate. But by sub- tracting them properly it will be found that their difference does not vanish. The method proposed to accomplish this subtraction will, in fact, give the combined effect of the self-action and mass correction of both diagrams II and III and II' and III' . It is based on the fact that an electron is never actually free. An electron's history will have always involved a series of scatterings, as will its future. These scatterings will be considered as oc- curring at long but finite time intervals. It will be sufficient to calculate the effect of self-action and the mass correction between any two of these scat- terings, since the result will evidently be the same between each pair of them. Then, the effect will be accounted for simply by regarding a correc- tion, equal to that calculated for one of the intervals between scatterings, as being associated with the potential at each scattering (number of inter- vals equals number of scatterings). Then, considering a single scattering event as here, this correction to the potential represents all the effects of diagrams H, III, II' , and III' . For an electron which is not quite free, p 2 * m 2 exactly, but instead p 2 = m 2 (l + e) 2 where 148 QUANTUM ELECTRODYNAMICS me = K/T by the uncertainty principle, and T is the interval between scatterings. Since T is large, e is a small quantity. Let $ = (1 + e)$ Q , where $ is the momentum of a free electron. If i. and # are the momentum representatives of the scattering poten- tials at a and b (any two scatterings), then the matrix of the amplitude to go from the initial state at a to the final state at b without any perturba- tions is p i - m * p 2 - m 2 ^ 2m 2 e up to terms of order €. With the perturbations of self-action and mass cor- rection, this matrix is ■, 2 fi/ I I 1 , d 4 k -A 2 (a) Without perturbation -Am (b) With perturbation of self-action and mass correction FIG. 29-2 It is the value of this matrix compared to that of the unperturbed matrix which gives the desired correction term (see Fig. 29-2). Problem: Show that for two noncommuting (or commuting) oper- ators A and B, the following expansion is true: 1 1 1^1 l^lr.1 = — - — B— + — B — B — + A + B A A A A A A Using the result of the preceding problem, one can write CORRECTION" TERMS 149 1 tf-tf-m rf + etf -\L-m tf -)£-m tf -)£-m 1 x e& K^ - \i - m so that the foregoing matrix becomes • a 2 f i; tf + m 1 ^ + m , d* d 4 k -A 2m z e 'M ^ -k-m ^ 2m z £ " k z -A 2 min k 2 - A 2 l47re J **2m 2 ^ ^ -K-m Po ^o-K-m^ 1?^?~ 1^ + m , / -A 2 \ u j± m . yf+m 2m 2 Mi?^^J-^ 2m 2 e Am f^7 i The first and last terms are identical, up to terms of order e, hence may be canceled. The integral in the second term has already been done essentially in computing diagram I (Fig. 28-2), except here ^ replaces &, $ it and $ 2 > so that <& = $2 ~ $1 ~ in this case and gives the result 2tt p 2m 2 e m 2m 2 * To this order in e the ^'s in the numerator may be replaced by $ 's. It is also noted that since ^ u = mu, # +m) A # + m) = 2m 2 $ + m) so that the foregoing result may be written This is just -(e 2 /27r)r times the matrix for no perturbation. Hence the cor- rection term due to diagrams II, III, II', and III' is obtained simply by re- placing the scattering potential i. by — (e 2 /27r)r^, as was stated earlier. It should be noted that the difficulty in obtaining the proper subtraction of the self-action and mass corrections just clarified does not represent a "divergence" problem of quantum electrodynamics. It is a typical problem which could as well arise in nonrelativistic quantum mechanics if, for ex- ample, one chose some nonzero value as a reference of potential, that is, regarded a free electron as moving in a uniform nonzero potential. It may be easily verified that this would give rise to an "energy correction" for the free electron analogous to the mass correction involved here. Then in 150 QUANTUM ELECTRODYNAMICS computing the amplitude for a scattering process where one used a "theo- retical energy" and subtracted the effect of the "energy correction," the difference of infinite terms would appear if one used free-electron wave functions. In this simple case the infinite term would, indeed, cancel upon proper subtraction but in principle the problem is the same as the present one. Finally, the complete correction term arising from self-action and mass correction is ei 2ir 2fto r *S--l)(l--^) + 8tane + -± \ A m in /\ tan 20/ tan tan 26 x 'o J a tan a da e 2 .,, ,,. 2d i + ^zziM- M) 87rm sin 26 RESOLUTION OF THE FICTITIOUS "INFRARED CATASTROPHE" From the correction term just determined, it is seen that, to order e 2 , the cross section for scattering of an electron with the emission of no pho- tons is ■"si- 1 t-A* f n 7 dent \on A m i n / where <7 is the cross section for the potential i. only. This cross section diverges logarithmically as A. m i n — * 0, and it is this divergence which was formerly referred to as the "infrared catastrophe." This result, however, arises from the physical fact that it is impossible to scatter an electron with the emission of no photons. When the electron is scattered, the electromagnetic field must change from that of a charge mov- ing with momentum pj to that for momentum p 2 . This change of the field is necessarily accompanied by radiation. In the theory of brehmsstrahlung, it was shown that the cross section for emission of one low-energy photon is 2 e_ d^jti f gPl e __ ^P2' e \ do; a ° 7T 47T \ Pf q P2*<1 / w Problem: Show that the integral over all directions and the sum over polarizations of the foregoing cross section is o- = cr Q (2e 2 A)[l - (20/tan 26)] duj/co where sin 2 6 = -($ 2 ~ F^i) 2 /4m 2 . Thus the probability of emitting any photon between k = and k = K^ is CORRECTION" TERMS 151 2e 7r V tan 20 A w ' a ° f \ tan 20/ n A min which diverges logarithmically. Therefore, the dilemma of the diverging scattering cross section actually arises from asking an improper question: What is the chance of scattering with the emission of no photons? Instead, one should ask: What is the chance of scattering with the emission of no photon of energy greater than K^ ? For there will always be some very soft photons emitted. Then, effectively, what is sought in answer to the last question is the chance of scattering and emitting no photon, the chance of emitting one pho- ton of energy below K^ , and the chance of two and more photons below K m (but these terms are of order e 4 and higher and hence are neglected). Each of these terms is infinite, actually, but is kept finite temporarily by the artifice of the X m i n . Their sum, however, does not diverge, as may be seen by gathering the previous results and by writing Chance of scattering and emitting no photon of energy > K 2 = *o i i ; - ! pendent of A. min of order e 4 ) 2e 2 / 20 \ . K m 1 - ~ ~ In ; + (terms 7T \ tan 20; A min ^o °b 2 \ / «„ \-i /terms independent \ 1 - S! 2 In -a- 1 - -2M1 + of X min and of { , K m A tan2 9 jj l orde »» 4 j This does not depend on X m i n and hence resolves the "infrared catastrophe." It has been shown by Bloch and Nordsieck that the same idea applies to all orders, t It is interesting that the largest term in the quantum-electrodynamic corrections to the scattering cross section, namely, -(2e 2 A) [1 - (20/tan 20)] In (m/K m ) may be obtained from classical electrodynamics, since such long wave- lengths are involved. The other terms have small effects. To date, the scat- tering experiments have been accurate enough to verify the existence of the large term but not accurate enough to verify the exact contributions of the smaller terms. Hence they do not provide a nontrivial test of quantum elec- trodynamics. These same considerations apply in any process involving the deflection fF. Bloch and A. Nordsieck, Phys. Rev., 52, 54 (1937) 152 QUANTUM ELECTRODYNAMICS of free electrons. The best way to handle the problem is to calculate every- thing in terms of the A. m j n and then to ask only questions which can have a sensible answer as verified by the eventual elimination of the A. m i n . Problem: Prepare diagrams and integrals needed for the radia- tive corrections (of order e 2 ) to the Klein-Nishina formula. Do as much as possible and compare results with those of L. Brown and R. P. Feynman.f Thirtieth Lecture ANOTHER APPROACH TO THE INFRARED DIFFICULTY Instead of introducing an artificial mass, assume no weak photons con- tribute. Thus we must subtract from the previous results the contributions of all photons with momentum magnitude less than some number k »\. The previous result is i{l + (e 2 /27r)[2 In (m/X min - 1)(1 -20/tan 29)] + 9 tan 9 r e + (4/tan 29) J q y tan y dyj} (30-1) The term to be subtracted is (e 2 /27r) J q °y u (p 2 - J£ + m)(k 2 -2p 2 ■ k^" 1 rftfi -tf+ m) x (k 2 - 2 Pl -k!)" 1 y^ d 4 k/(k 2 -A 2 min ) (30-2) We assume k « Pi or p 2 , and neglect both K and the first two k 2 in this integral. Then using ^y^ = 2p^ - y^i, the integral is approximately -|2 P2 M _ JBlt <Li f 27T 2 J p 2 • k Pi * k. d 4 k k 2 - X 2 • (30-3) Then x = e 2 /27r{[l - (20/tan 2 9)] [2 In (2koA min - 1)] + [40/tan 29] r 26 x [(1/20) J o (y/tan y) dy - 1]} (30-4) This is the term to be subtracted from expression (30-1). Using sin 2 = q 2 /4m 2 , for small q, Eq. (30-4) becomes x = (e 2 /27r) (2q 2 /3m 2 )[ln (2k A min ) - (5/6)] tPhys. Rev., 85, 231 (1952). CORRECTION" TERMS 153 Subtracting this from Eq. (30-1), also with q small, gives i {1 + (e 2 /47r)(4q 2 /3m 2 )[ln M/A min ) - (3/8) - ln(2koAmin) + (5/6)]} (30-5) The last term is [In (M/2ko) + (11/24)]. EFFECT ON AN ATOMIC ELECTRON Consider the hydrogen atom with a potential V = e 2 /r and a wave func- tion 4>q(B.) exp (-iE t) = o (x fI ). Take the wave function to be normalized in the conventional manner. The effect of the self-energy of the electron is to shift the energy level by an amount AE = e 2 /0o(x2 J t 2 )r /i K + v (2,l)r f ,6 + (s 1) 2 2 )0o(xi,t 1 )d 3 x 1 d 3 x 2 dt 2 -Am/ 0(x,t)0(x,t) d l (30-6) The first integral is written down from Fig. 30-1. The second is the free- particle effect as noted in previous lectures. The kernel K + v is not well FIG. 30-1 enough determined to make exact calculation of this integral possible. An approximate calculation can be made with the form K + V (2,l) = £ expI-iEoflfc-tjj] ? n (x 2 )0 n (x 1 ) t 2 >-tj - similar sum over negative energies for t 2 < t t The photon propagation kernel can be expanded as 154 QUANTUM ELECTRODYNAMICS <5+(si )2 2 ) = 4tt Jexp[-ik(t 2 -t 1 ) + ik(x 2 -x 1 )] d 3 k/2k(27r)~ 3 t 2 >ti = 4?r j exp [+ ik(t 2 - t t ) + ik(x 2 -x^l d 3 k/2k(27r)~ 3 t 2 <tj Using these expressions, Eq. (30-6) becomes AE = S/[Q! M exp(-iK-R)] 0n (E n +K-E )- 1 [a fi exp(iK'R)] n0 + n x d 3 k/47rk _ £ J[a u exp(-iK-R)] 0n (|E n |+ u+ Eq)" 1 -n x [a^ exp (iK • R)] n0 d 3 k/47rk - (Am term) (30-7) This form implies the use of <fi* instead of and a 4 = 1, a lj2j3 = a. Another approach to the motion of an electron in a hydrogen atom is the following. Consider the electron as a free particle intermittently scattered by the Coulomb potential. The scatterings cause a phase shift in the wave function of the order of (Rydberg/R ). Thus the period between scatterings is of the order T = K/Rydberg. Take the lower limit k of the momentum of the "self-action" photons as very large compared to the Rydberg. Then it is very probable that an emitted photon will be reabsorbed before two inter- actions between the electron and the potential have taken place; it is very improbable for two or more scatterings to take place between emission and absorption (see Fig. 30-2). Then the correction to the potential is that com- puted in Eq. (30-5) for small q (plus anomalous moment correction). This is (e 2 /47r)(4q 2 /3m 2 )(ln m/2k + 11/24) f in momentum space. To transform to ordinary space, use q 2 V = (q 4 2 - Q 2 ) t — (9 2 /9t 2 - V 2 ) V Thus the correction is -(e 2 /37rm 2 )(log M/2k + 11/24) V 2 V (30-7') This correction is of greatest importance for the s state, since with a Cou- lomb potential V 2 V = 47rZe 2 S(R), and only in the s states is 0(R) different from at R= 0. The choice of ko is determined by the inequalities m »k » Rydberg. A satisfactory value is k = 137 Ryd. With such a k , the effect of photons of k < k must be included. This will be done by separating the effect into the sum of three contributing effects. It will be seen that two of these effects CORRECTION" TERMS 155 k » Rydberg probable improbable FIG. 30-2 are independent of the potential V and thus are canceled by similar terms in the Am correction for a free particle. Thus for only one situation must the effect be computed. In all cases, since k is small, the nonrelativistic approximation to expression (30-7) may be used. (1) The contribution of negative energy states: Neglecting k with respect to m gives (|E n | +k+ E ) « 2m The matrix element for a 4 is very small, and only the elements for a need be considered. Then the sum over negative states is S/W«0n)' (O!n0)/2m]k 2 dk/k If this sum is continued for +n, a negligible term of order v 2 /c 2 is added. Thus the sum is approximately "Si t(«0n) ' Ko)/2m] k 2 dk/k = (a ■ a) 00 k 2 dk/2mk all states = 3k 2 /4m This is independent of V, and thus is canceled by a similar quantity in the Am term. (2) Longitudinal positive energy states (a u —* Ot • k/k): As an exercise the reader may show a • k exp (ik • R) = H exp (ik ■ R) - exp (ik ■ R)H 156 QUANTUM ELECTRODYNAMICS Then [(a • k/k) exp (ik • R)] n0 = (E n - E )/k[exp (ik • R)] n0 and the contribution of these terms summed over positive energy states gives / [1 - (E n - E ) 2 /k 2 ] exp (ik • R) 0n exp (-ik • R) n0 (E n + k - Eq)' 1 d 3 k/47rk = J(E n - E + k) exp (ik • R) 0n exp (-ik ■ R) n0 d 3 k/47rk 3 = /[H exp (ik • R) - exp (ik • R)H] 0n [exp - (ik • R)] n0 d 3 k/47rk 3 Writing H = p 2 /2m (V commutes with the exponent), this becomes f[(p + k) 2 /2m - p 2 /2m + k] d 3 k/k 3 This term is independent of V, and thus is also canceled by the Am correc- tion. (3) Transverse positive energy states: Since k is large compared to the size of the atom, the dipole approximation can be used.f The general term in the sum of Eq. (30-7) becomes J(a tT ) 0n (a tT ) n0 (E n + k - Eq)" 1 d 3 k/k (30-8) Writing (E^k-Eo)- 1 ^ 1/k- (E n -E )/(E n +k-E )k the term in 1/k can be split off from the rest of the integral as a quantity independent of V and thus canceled by the Am correction. Further, by averaging over directions, Kr)0n (Oitr.)n0 = 2/3(a) 0n ' (O) n0 = (2/3m 2 )(p) 0n ' (p) n0 in the nonrelativistic approximation. Thus the integral of Eq. (30-8) is (2/3m 2 )(p) 0n • (p) n0 (E n - E ) log (k + E n - E )/(E n - E ) Using the relation Pn o(E n -E ) = (pH-Hp) n0 = (VV) n0 |Cf. H. Bethe, Phys. Rev., 72, 339 (1947), "CORRECTION" TERMS 157 and the fact that k$ » E n - E , one part of the sum over transverse positive energy states is In k £Pon- (VV) n0 = 1/2 In k (V 2 V) o n This cancels with the In ko of Eq.(30-7'), leaving the final correction as (2e 2 /37rm 2 ) £ p n0 'Pon^ - E ){ log [M/2(E n - E )] + (11/24)} + n + anomalous moment correction This sum has been carried out numerically to be compared with the observed Lamb shift. Thirty -first Lecture CLOSED-LOOP PROCESSES, VACUUM POLARIZATION Another process which is still of first order in e 2 has not been consid - ered in the scattering by a potential. Instead of the potential scattering the particle directly, it can do so by first creating a pair which subsequently annihilates, creating a photon which does the scattering. Diagram I (Fig. 31-1) applies to this process; diagram II applies to a similar process, with the order in time changed slightly. The amplitude for these processes is spin states ** \ *• n. i ofu (31_1) where u is the spinor part of the closed-loop wave function. The first pa- renthesis is the amplitude for the electron to be scattered by the photon; 1/q 2 is the photon propagation factor; and the second parenthesis is the am- plitude for the closed-loop process which produces the photon. The expres- sion is integrated over p because the amplitude for a positron of any mo- menta is desired. In the sum over four spin states of u, two states take care of the processes of diagram I and two states take care of the proc- esses of diagram II. No projection operators are required, so the method of spurs may be used directly to give i47re 2 (uzT^Uj)-? /s P 1 7u q Lp-m n tf+d-m -4 j4 (27T)- 4 d 4 p (31-2) a form which contains both I and II (so as usual it is not necessary to make separate diagrams for processes whose only difference is the order in time). 158 QUANTUM ELECTRODYNAMICS This integral also diverges, but a photon convergence factor, as used in the previous lectures, is of no value because now the integral is over p, the mo- mentum of the positron in the intermediate step. The method which has been used to circumvent the divergence difficulty is to subtract from this integral, a similar integral with m replaced by M. M is taken to be much larger closed loop f*2 = tfi + i closed loop i>2 = i>\~& than m, and this results in a type of cutoff in the integral over p. When this is done, the amplitude is found to be f (u 2 ynUi)a u (e 2 A)[-(l/3)ln(M/m) 2 - (1 - 0/tan 9) x (4m 2 + 2q 2 )/3q 2 + 1/9] (31-3) tSee R. P. Feynman, Phys. Rev., 76, 769 (1949); included in this volume. 1 'CORRECTION" TERMS 159 where q 2 = 4m 2 sin 2 6, which, for small q, becomes (Z 2 y ll u i )a ll (e 2 /7T)[-(l/3)ln(M/m) 2 + 2q 2 /l5] (31-4) Notice that (u 2 y JL( u 1 ) = (u 2 ^u 1 ), so that, considering only the divergent part of the correction, the effective potential is i {1 + (e 2 /7T)[- (1/3) In (M/m) 2 ]} (31-5) The 1 comes from the theory without radiative corrections, while the e 2 term is the correction due to processes of the type just described. Thus the correction can be interpreted as a small reduction in the effect of all potentials, and one can introduce an experimental charge e exp and a theo- retical charge e^ related by eexp^th+Ae (31-6) where A(e 2 ) = -(e 2 /37r) In (M/m) 2 , in a manner analogous to the mass cor- rection described in Lecture 28. This is referred to as "charge renormal- ization." The other term, (2/15)(e 2 /7T)qV is more interesting, since it represents a perturbation 2e 2 /157r (V 2 V). This correction is responsible for 27 Mc in the Lamb shift and the {in [m/2(E n -E )] + (11/24)} term in (30-7') is replaced by {in [m/2(E n - E )] + (11/24) -(1/5)}. The 1/5 term is due to the "polarization of the vacuum." SCATTERING OF LIGHT BY A POTENTIAL One possible process for the scattering of light, and an indistinguishable alternative, is indicated by the diagrams in Fig. 31-2. The second diagram differs from the first only in the direction of the arrows of the electron lines. Reversing such a direction is equivalent to changing an electron to a posi- tron. Thus the coupling with each potential would change sign. Since there are three such couplings, the amplitude for the second process is the nega- tive of that for the first. Since the amplitudes add, the net amplitude is zero. In general, any closed-loop process of this type involving an odd number of couplings to a potential (including photon), has zero net amplitude. Problem: Set up the integrals for each of the two diagrams in Fig. 31-2 and show that they are equal and opposite in sign. However, the higher-order processes shown in Fig. 31-3 can take place. The amplitude for the process is 160 QUANTUM ELECTRODYNAMICS K 2 FIG. 31-2 -til f* ^ 'til alternatives V2 FIG. 31-3 < 'CORRECTION " TERMS 161 -(47re 2 ) 2 /Sp WtdJ - m)" 1 fafa -fa - m)" 1 ^ -fa-fa- m)" 1 x^ 4 (^ + *-m)- 1 ] (27r)" 4 d 4 k plus five similar terms resulting from permuting the order of photons. This integral appears to diverge logarithmically. But when all six alternatives are taken into account, the sum leaves no divergent term. More complicated closed-loop processes are convergent. Pauli Principle and the Dirac Equation In Lecture 24 the probability of a vacuum remaining a vacuum under the influence of a potential was calculated. The potential can create and anni- hilate pairs (a closed-loop process) between times tj and t 2 . The amplitude for the creation and annihilation of one pair is (to first nonvanishing order) L ~ //sp [K + (l,2)^(2)K + (2, l)rf(l)] dn dr 2 The amplitude for the creation and annihilation for two pairs is a factor L for each, but, to avoid counting each twice when integrating over all dri and dT2, it is L 2 /2. For three pairs the amplitude is L 3 /3!. The total amplitude for a vacuum to remain a vacuum is, then, c v = 1 - L+ L 2 /2! - L 3 /3! + -" = e" L (31-7) where the 1 comes from the amplitude to remain a vacuum with nothing happening. The use of minus signs for the amplitude for an odd number of pairs can be given the following justification in terms of the Pauli principle. Suppose the diagram for t < t t is as shown in Fig. 31-4. The completion of this process can occur in two ways, however (see Fig. 31-5). The second way can be thought of as obtained by the interchange of the two electrons, hence the amplitude of the second must be subtracted from that of the first, FIG. 31-4 162 PAULI PRINCIPLE AND DIRAC EQUATIONS 163 FIG. 31-5 according to the Pauli principle. But the second process is a one -loop proc- ess, whereas the first process is a two-loop process, so it can be concluded that amplitudes for an odd number of loops must be subtracted. The prob- ability for a vacuum to remain a vacuum is vac- vac |c v | 2 = exp (-2 real part of L) The real part of L (R.P. of L) may be shown to be positive, so it is clear that terms of the series must alternate in sign in order that this probability be not greater than unity. We have, therefore, two arguments as to why the expression must be e" L . One involves the sign of the real part, a property just of K + and the Dirac equation. The second involves the Pauli principle. We see, therefore, that it could not be consistent to interpret the Dirac equation as we do un- less the electrons obey Fermi-Dirac statistics. There is, therefore, some connection between the relativistic Dirac equation and the exclusion princi- ple. Pauli has given a more elaborate proof of the necessity for the exclu- sion principle but this argument makes it plausible. This question of the connection between the exclusion principle and the Dirac equation is so interesting that we shall try to give another argument that does not involve closed loops. We shall prove that it is inconsistent to assume that electrons are completely independent and wave functions for several electrons are simply products of individual wave functions (even though we neglect their interaction). For if we assume this, then Probability of vacuum remaining a vacuum Probability of vacuum to 1 pair Probability of vacuum to 2 pairs PV E iKlpairl 2 all pairs P V E l K lpair| 2 iKlpairl 2 all pairs 164 QUANTUM ELECTRODYNAMICS Now, the sum of these probabilities is the probability of a vacuum becoming any thing and this must be unity. Thus 1 = P v [1 + (prob. of 1 pair) + (prob. of 2 pairs) + • • •] (31-8) The probability that an electron goes from a to b and that nothing else hap- pens is P V |K + (b,a)| 2 . The probability that the electron goes from a to b and one pair is produced is P v |K + (b,a)| 2 |K(1 pair)| 2 , and the probability that the electron goes from a to b with two pairs produced is P v |K + (b,a)| 2 x |K(2 pair)| 2 . Thus the probability for an electron to go from a to b with any number of pairs produced is P v |K + (b,a)| 2 [l + |K(lpair)| 2 + |K(2pairs)| 2 + . ■ . = |K + (b,a)| 2 (31-9) [see Eq. (31-8)] . Now since the electron must go somewhere, J|K + (b,a)| 2 db=l However, it is a property of the Dirac kernel that /|K + (b, a)| 2 db> 1 (31-10) and an inconsistency results. The inconsistency can be eliminated by assum- ing that electrons obey Fermi-Dirac statistics and are not independent. Un- der these circumstances the original electron and the electron of the pair are not independent and Probability of electron from , 4. u i i • ^ j r < K+(b,a)r K(l pair) ' a to b plus 1 pair produced ' (31-11) because we should not allow the case that the electron in the pair is in the same state as the electron at b. For the kernel of the Klein-Gordon equation, it turns out that the sign of the inequality in Eq. (31-10) is reversed. Therefore, for a spin-zero parti- cle neither Fermi-Dirac statistics nor independent particles are possible. If the wave functions are taken symmetric (charges reversed add ampli- tudes, Einstein-Bose statistics), the inequality Eq. (31-11) is also reversed. In symmetrical statistics the presence of a particle in a state (say 6) en- hances the chance that another is created in the same state. So the Klein- Gordon equation requires Bose statistics. It would be interesting to try to sharpen these arguments to show that the difference between J |K + (b,a)| 2 db and 1 is quantitatively exactly compen- sated for by the exclusion principle. Such a fundamental relation ought to have a clear and simple exposition. 165 AN OPERATOR CALCULUS 123 10. SUMMARY OF NUMERICAL FACTORS FOR TRANSITION PROBABILITIES The exact values of the numerical factors appearing in the rules of II for computing transition probabilities are not clearly stated there, so we give a brief summary here. 20 The probability of transition per second from an initial state of energy £ to a final state of the same total energy (assumed to be in a continuum) is given by (h=c=l), Prob. trans/sec = 27riV- 1 |3H| 2 p(£), where p(E) is the density of final states per unit energy range at energy E and 1 9TI | 2 is the square of the matrix element taken between the initial and final state of the transition matrix 9TC appropriate to the problem. N is a normalizing constant. For bound states conventionally normalized it is 1. For free particle states it is a product of a factor Ni for each particle in the initial and for each in the final energy state. Ni depends on the normalization of the wave functions of the particles (photons are considered as particles) which is used in computing the matrix element of 2(11. The simplest rule (which does not destroy the apparent covariance of 911), is 21 A r ,= 2ei, where e,- is the energy of the particle. This corresponds to choosing in momentum space, plane waves for photons of unit vector potential, e 2 = — 1. For electrons it corresponds to using {uu) = 2m (so that, for example, if an electron is deviated from initial ^ t to final p 2 , the sum over all initial and final spin states of |3TZ| 2 is Sp[(p t +m)m(pi+m)ml). Choice of norma- lization (uy t u)=?l results in A r ,= l for electrons. The matrix 311 is evaluated by making the diagrams and following the rules of II, but with the following defini- tion of numerical factors. (We give them here for the special case that the initial, final, and intermediate 20 In I and II the unfortunate convention was made that d*k means dk i dkidk 2 dk i (2Tr)~ 2 for momentum space integrals. The confusing factor (2*)^ here serves no useful purpose, so the con- vention will be abandoned. In this section d*k has its usual meaning, dk<dkydk*lk 3 . states consist of free particles. The momentum space representation is then most convenient.) First, write down the matrix directly without numerical factors. Thus, electron propagation factor is (p—m)~ l , virtual photon factor is k~ 2 with couplings TV • "y> A real photon of polarization vector e M con- tributes factor e. A potential (times the electron charge, e) A^(x) contributes momentum q with amplitude a(q), where a»{q) = fA^X) exp(iq-xi)d 4 xi. (Note: On this point we deviate from the definition of a in I which is there (27r)~ 2 times as large.) A spur is taken on the matrices of a closed loop. Because of the Pauli principle the sign is altered on contributions corresponding to an exchange of electron identity, and for each closed loop. One multiplies by {2ir)^d i p={2ir)~ i dp l dp J (lp v dp z and integrates over all values of any undetermined mo- mentum variable p. (Note: On this point we again differ. 20 ) The correct numerical value of 91X is then obtained by multiplication by the following factors. (1) A factor (4ir)*e for each coupling of an electron to a photon. Thus, a virtual photon, having two such couplings, contributes 47re 2 . (In the units here, e 2 = 1/137 approxi- mately and (47r)*e is just the charge on an electron in heaviside units.) (2) A further factor —i for each virtual photon. For meson theories the changes discussed in II, Sec. 10 are made in writing 2flZ, then further factors are (1) (47r)*g for each meson-nucleon coupling and (2) a factor —i for each virtual spin one meson, but -\-i for each virtual spin zero meson. This suffices for transition probabilities, in which only the absolute square of 9TC is required. To get 311 to be the actual phase shift per unit volume and time, additional factors of i for each virtual electron propa- gation, and — i for each potential or photon interaction, are necessary. Then, for energy perturbation problems the energy shift is the expected value of i3Tl for the unperturbed state in question divided by the normal- ization constant Ni belonging to each particle compris- ing the unperturbed state. The author has profited from discussions with M. Peshkin and L. Brown. 21 In general, A, is the particle density. It is Ni=(uytu) for spin one-half fields and i[(<j>*d<j>/dt)-<t>d<j>*/dtl for scalar fields. The latter is 2a if the field amplitude <t> is taken as unity. 167 PHYSICAL REVIEW VOLUME 76 MBER 6 SEPTEMBER 15 The Theory of Positrons R. P. Feynman Department of Physics, Cornell University, Ithaca, New York (Received April 8, 1949) The problem of the behavior of positrons and electrons in given external potentials, neglecting their mutual interaction, is analyzed by replacing the theory of holes by a reinterpretation of the. solu- tions of the Dirac equation. It is possible to write down a complete solution of the problem in terms of boundary conditions on the wave function, and this solution contains automatically all the possibilities of virtual (and real) pair formation and annihilation together with the ordinary scattering processes, including the correct relative signs of the various terms. In this solution, the "negative energy states" appear in a form which may be pictured (as by Stuckelberg) in space-time as waves traveling away from the external potential backwards in time. Experimentally, such a wave corresponds to a positron approach- ing the potential and annihilating the electron. A particle moving forward in time (electron) in a potential may be scattered forward in time (ordinary scattering) or backward (pair annihilation). When moving backward (positron) it may be scattered backward in time (positron scattering) or forward (pair production). For such a particle the amplitude for transition from an initial to a final state is analyzed to any order in the potential by considering it to undergo a sequence of such scatterings. The amplitude for a process involving many such particles is the product of the transition amplitudes for each particle. The exclusion principle requires that antisymmetric combinations of amplitudes be chosen for those complete processes which differ only by exchange of particles. It seems that a consistent interpre- tation is only possible if the exclusion principle is adopted. The exclusion principle need not be taken into account in intermediate states. Vacuum problems do not arise for charges which do not interact with one another, but these are analyzed nevertheless in anticipation of application to quantum electrodynamics. The results are also expressed in momentum-energy variables. Equivalence to the second quantization theory of holes is proved in an appendix. 1. INTRODUCTION THIS is the first of a set of papers dealing with the solution of problems in quantum electrodynamics. The main principle is to deal directly with the solutions to the Hamiltonian differential equations rather than with these equations themselves. Here we treat simply the motion of electrons and positrons in given external potentials. In a second paper we consider the interactions of these particles, that is, quantum electrodynamics. The problem of charges in a fixed potential is usually treated by the method of second quantization of the electron field, using the ideas of the theory of holes. Instead we show that by a suitable choice and inter- pretation of the solutions of Dirac's equation the prob- lem may be equally well treated in a manner which is fundamentally no more complicated than Schrodinger's method of dealing with one or more particles. The vari- ous creation and annihilation operators in the conven- tional electron field view are required because the number of particles is not conserved, i.e., pairs may be created or destroyed. On the other hand charge is conserved which suggests that if we follow the charge, not the particle, the results can be simplified. In the approximation of classical relativistic theory the creation of an electron pair (electron A , positron B) might be represented by the start of two world lines from the point of creation, 1. The world lines of the positron will then continue until it annihilates another electron, C, at a world point 2. Between the times h and t 2 there are then three world lines, before and after only one. However, the world lines of C, B, and A together form one continuous line albeit the "positron part" B of this continuous line is directed backwards in time. Following the charge rather than the particles corresponds to considering this continuous world line as a whole rather than breaking it up into its pieces. It is as though a bombardier flying low over a road suddenly sees three roads and it is only when two of them come together and disappear again that he realizes that he has simply passed over a long switchback in a single road. This over-all space-time point of view leads to con- siderable simplification in many problems. One can take into account at the same time processes which ordi- narily would have to be considered separately. For example, when considering the scattering of an electron by a potential one automatically takes into account the effects of virtual pair productions. The same equation, Dirac's, which describes the deflection of the world line of an electron in a field, can also describe the deflection (and in just as simple a manner") when it is large enough to reverse the time-sense of the world line, and thereby correspond to pair annihilation. Quantum mechanically the direction of the world lines is replaced by the direction of propagation of waves. This view is quite different from that of the Hamil- tonian method which considers the future as developing continuously from out of the past. Here we imagine the entire space-time history laid out, and that we just become aware of increasing portions of it successively. In a scattering problem this over-all view of the com- plete scattering process is similar to the 5-matrix view- point of Heisenberg. The temporal order of events dur- ing the scattering, which is analyzed in such detail by the Hamiltonian differential equation, is irrelevant. The relation of these viewpoints will be discussed much more fully in the introduction to the second paper, in which the more complicated interactions are analyzed. The development stemmed from the idea that in non- relativistic quantum mechanics the amplitude for a given process can be considered as the sum of an ampli- 749 168 750 R. P. FE YN MAN tude for each space-time path available. 1 In view of the fact that in classical physics positrons could be viewed as electrons proceeding along world lines toward the past (reference 7) the attempt was made to remove, in the relativistic case, the restriction that the paths must proceed always in one direction in time. It was dis- covered that the results could be even more easily understood from a more familiar physical viewpoint, that of scattered waves. This viewpoint is the one used in this paper. After the equations were worked out physically the proof of the equivalence to the second quantization theory was found. 2 First we discuss the relation of the Hamiltonian differential equation to its solution, using for an example the Schrodinger equation. Next we deal in an analogous way with the Dirac equation and show how the solu- tions may be interpreted to apply to positrons. The interpretation seems not to be consistent unless the electrons obey the exclusion principle. (Charges obeying the Klein-Gordon equations can be described in an analogous manner, but here consistency apparently requires Bose statistics.) 3 A representation in momen- tum and energy variables which is useful for the calcu- lation of matrix elements is described. A proof of the equivalence of the method to the theory of holes in second quantization is given in the Appendix. 2. GREEN'S FUNCTION TREATMENT OF SCHRO DINGER'S EQUATION We begin by a brief discussion of the relation of the non-relativistic wave equation to its solution. The ideas will then be extended to relativistic particles, satisfying Dirac's equation, and finally in the succeeding paper to interacting relativistic particles, that is, quantum electrodynamics. The Schrodinger equation id$/dt = H$, (1) describes the change in the wave function \f/ in an infinitesimal time At as due to the operation of an operator exp( — iHAt). One can ask also, if ^(x x , h) is the wave function at Xi at time t\, what is the wave function at time t 2 >ti? It can always be written as / iA(x 2 , h) = K(x it h; xi, *i)iKxi, h)dhL X (2) where if is a Green's function for the linear Eq. (1). (We have limited ourselves to a single particle of co- ordinate x, but the equations are obviously of greater generality.) If H is a constant operator having eigen- values E„, eigenfunctions 4> n so that ^(x, t\) can be ex- panded as £n C„0 n (x), then ^(x, / 2 ) = exp(— iE n {t 2 — h)) XC„</>„(x). Since C n = f<j> n * {^1)^1, h)d z x h one finds 1 R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948). 'The equivalence of the entire procedure (including photon interactions) with the work of Schwinger and Tomonaga has been demonstrated by F. J. Dyson, Phys. Rev. 75, 486 (1949). 3 These are special examples of the general relation of spin and statistics deduced by W. Pauli, Phys. Rev. 58, 716 (1940). (where we write 1 for Xi, h and 2 for x 2 , t 2 ) in this case K(2, l) = Z 0»(x 2 )«»*(x 1 ) exp(-iE n (t 2 -h)), (3) for ta>k- We shall find it convenient for h<h to define K{2, 1) = (Eq. (2) is then not valid for t 2 <h). It is then readily shown that in general K can be defined by that solution of (id/dt 2 -H 2 )K(2,l) = id(2,l), (4) which is zero for t 2 <t h where 5(2, l) = d(l 2 — h)8(x 2 — x{) X5(y 2 —yi)5(z 2 —Zi) and the subscript 2 on H 2 means that the operator acts on the variables of 2 of if (2, 1). When H is not constant, (2) and (4) are valid but K is less easy to evaluate than (3). 4 We can call K{2, 1) the total amplitude for arrival at x 2 , t 2 starting from Xi, t\. (It results from adding an amplitude, expz'S, for each space time path between these points, where 5 is the action along the path. 1 ) The •transition amplitude for finding a particle in state x(x 2 , ti) at time t 2 , if at h it was in ^(xi, ti), is / X*(2)K(2, l)^(l)d 3 x l( f 3 x,. (5) A quantum mechanical system is described equally well by specifying the function K, or by specifying the Hamiltonian H from which it results. For some purposes the specification in terms of K is easier to use and visualize. We desire eventually to discuss quantum electrodynamics from this point of view. To gain a greater familiarity with the K function and the point of view it suggests, we consider a simple perturbation problem. Imagine we have a particle in a weak potential U(x, t), a function of position and time. We wish to calculate K{2, 1) if U differs from zero only for t between /1 and t 2 . We shall expand K in increasing powers of U: K{2, l) = K (2, l)+if (1) (2, l)+if< 2 >(2, 1)+ • • -. (6) To zero order in U, K is that for a free particle, K (2, l). 4 To study the first order correction if (1) (2, 1), first con- sider the case that U differs from zero only for the infinitesimal time interval Ah between some time h and/3+A/ 3 (^i<^3<^). Then if ^(1) is the wave function at xi, h, the wave function at x 3 , t z is :/ *(3)= A'o(3, \)+{l)dH u (7) since from ti to / 3 the particle is free. For the short interval A/ 3 we solve (1) as ^(x, /■H-A/ 3 ) = exp(-;#A/ 3 )^(x, / 3 ) = (l-iH Ah-iUAhU(x, / 3 ), 4 For a non-relativistic free particle, where 4>„ = exp(/p-x), E n = p 2 /2m, (3) gives, as is well known K (2, l)=Jexp[-(Jp-x l -»p-x 2 )-ip 2 (/ 2 -^)/2w]^P(2T) -3 = (27rim- 1 ('2-ii)) _§ exp(^'w(x 2 -x,) 2 (^-/ 1 )- 1 ) for h>h, and iT = for t t <h. 169 THEORY OF POSITRONS 751 where we put H = H -\-U, H being the Hamiltonian of a free particle. Thus i^(x, /3+A/3) differs from what it would be if the potential were zero (namely (l — iH Ah)\p(x, tz)) by the extra piece At = - iU(x h h) ■ $(x z , h) A/ 3 , (8) which we shall call the amplitude scattered by the potential. The wave function at 2 is given by i£(x 2 , k) ■■ ■ I #o(x 2) h; x 3 , t 3 +M z )f(xz, h+Ah)d 3 x 3 , since after /3+A/3 the particle is again free. Therefore the change in the wave function at 2 brought about by the potential is (substitute (7) into (8) and (8) into the equation for iA(x 2 , / 2 )): A^(2) = — i Jk (2, 3)U(3)K (3, 1)^(1)^x^X3^3. In the case that the potential exists for an extended time, it may be looked upon as a sum of effects from each interval A/ 3 so that the total effect is obtained by integrating over / 3 as well as x 3 . From the definition (2) of K then, we find KU(2, l)=-i Jk (2, 3)U(3)K (3,l)dr 3 , (9) where the integral can now be extended over all space and time, rfr 3 = d 3 x 3 d(z. Automatically there will be no contribution if / 3 is outside the range h to t 2 because of our definition, K (2, 1) = for h<h- We can understand the result (6), (9) this way. We can imagine that a particle travels as a free particle from point to point, but is scattered by the potential U. Thus the total amplitude for arrival at 2 from 1 can be considered as the sum of the amplitudes for various alternative routes. It may go directly from 1 to 2 (amplitude K (2, 1), giving the zero order term in (6)). Or (see Fig. 1(a)) it may go from 1 to 3 (amplitude K (3, 1)), get scattered there by the potential (scatter- ing amplitude —iU(3) per unit volume and time) and then go from 3 to 2 (amplitude K (2, 3)). This may occur for any point 3 so that summing over these alternatives gives (9). Again, it may be scattered twice by the potential (Fig. 1(b)). It goes from 1 to 3 (K (3, 1)), gets scattered there ( — iU(3)) then proceeds to some other point, 4, in space time (amplitude K (4, 3)) is scattered again (— iU(4)) and then proceeds to 2 (K (2, 4)). Summing over all possible places and times for 3, 4 find that the second order contribution to the total amplitude KW(2, 1) is (-i) 2 ffK (2,4)U(4)K (4,3) XU(3)K (3,l)dT,dT i . (10) This can be readily verified directly from (1) just as (9) (a) FIRST 0RDER,EQ(9) (b) SECOND ORDER. EQ.(IO) Fig. 1. The Schrodinger (and Dirac) equation can be visualized as describing the fact that plane waves are scattered successively by a potential. Figure 1 (a) illustrates the situation in first order. K (2, 3) is the amplitude for a free particle starting at point 3 to arrive at 2. The shaded region indicates the presence of the potential A which scatters at 3 with amplitude — iA(3) per cm 3 sec. (Eq. (9)). In (b) is illustrated the second order process (Eq. (10)), the waves scattered at 3 are scattered again at 4. How- ever, in Dirac one-electron theory iC (4, 3) would represent elec- trons both of positive and of negative energies proceeding from 3 to 4. This is remedied by choosing a different scattering kernel iP + (4, 3), Fig. 2. was. One can in this way obviously write down any of the terms of the expansion (6). 5 3. TREATMENT OF THE DIRAC EQUATION We shall now extend the method of the last section to apply to the Dirac equation. All that would seem to be necessary in the previous equations is to consider H as the Dirac Hamiltonian, ^ as a symbol with four indices (for each particle). Then K can still be defined by (3) or (4) and is now a 4-4 matrix which operating on the initial wave function, gives the final wave func- tion. In (10), U(3) can be generalized to A 4 (3) — a-A(3) where A 4, A are the scalar and vector potential (times e, the electron charge) and a are Dirac matrices. To discuss this we shall define a convenient rela- tivistic notation. We represent four-vectors like x, t by a symbol x„, where \i= 1, 2, 3, 4 and x t = t is real. Thus the vector and scalar potential (times e) A, A t is A?. The four matrices /3a, /3 can be considered as transform- ing as a four vector 7^ (our y m differs from Pauli's by a factor i for n= 1, 2, 3). We use the summation conven- tion a li b^ = a i b i — a-ibi— ajbi— a 3 6 3 =a-6. In particular if a M is any four vector (but not a matrix) we write a = a li y li so that a is a matrix associated with a vector (a will often be used in place of <z M as a symbol for the vector). The 7„ satisfy 7 M 7,.+ 7,7„= 25„„ where 5 4 4= + 1, 5n= 5 2 2= 5 3 3= — 1, and the other 5„„ are zero. As a consequence of our summation convention b liV a,=a ll . and 5^=4. Note that ab+ba=2a-b and that a^^a^a,, — a- a is a pure number. The symbol d/dx? will mean d/dt for ii = 4, and -d/dx, -d/dy, -d/dz for M =l, 2, 3. Call V=y fi d/dx ll =pd/dt+l3a-V. We shall imagine 6 We are simply solving by successive approximations an integral equation (deducible directly from (1) with H = H +U and (4) with H = H ), U2)=-ifK (2, 3)U(3)+(3)dr 3 +fKo(2, 1)^(1)^,, where the first integral extends over all space and all times *3 greater than the ti appearing in the second term, and h>h. 170 752 R. P. FE YNM AN K (3,3),POS. E. JNTAtoS ONLY NEG.E (b) VIRTUAL SCATTERING (c) VIRTUAL PAIR «4>»3 t 4 <t 3 SECOND ORDER, EO. (14) Fig. 2. The Dirac equation permits another solution K + {2, 1) if one considers that waves scattered by the potential can proceed backwards in time as in Fig. 2 (a). This is interpreted in the second order processes (b), (c), by noting that there is now the possi- bility (c) of virtual pair production at 4, the positron going to 3 to be annihilated. This can be pictured as similar to ordinary scattering (b) except that the electron is scattered backwards in time from 3 to 4. The waves scattered from 3 to 2' in (a) represent the possibility of a positron arriving at 3 from 2' and annihilating the electron from 1. This view is proved equivalent to hole theory: electrons traveling backwards in time are recognized as positrons. hereafter, purely for relativistic convenience, that <j> n * in (3) is replaced by its adjoint 4> n =4> n *fi- Thus the Dirac equation for a particle, mass m, in an external field A = A ll y lt is (iV-m)il/=A$, (11) and Eq. (4) determining the propagation of a free particle becomes (iV 2 -m)K + (2,l) = id(2,l), (12) the index 2 on V 2 indicating differentiation with respect to the coordinates x 2li which are represented as 2 in K+(2, 1) and 5(2, 1). The function K+(2, 1) is defined in the absence of a field. If a potential A is acting a similar function, say K+ (A) {2, 1) can be defined. It differs from K+(2, 1) by a first order correction given by the analogue of (9) namely #+ ci) (2, i: ij K+ (2, 3)A(3)K+(3, l)dn, (13) representing the amplitude to go from 1 to 3 as a free particle, get scattered there by the potential (now the matrix A(3) instead of U{3)) and continue to 2 as free. The second order correction, analogous to (10) is # + < 2 >(2,l)=-JJV + (2,4)>i(4) XA%(4, 3)A(3)K + (3, l)d u d u , (14) and so on. In general K+S A) satisfies {iV 2 -A(2)-m)K + ^{2, l) = id(2, 1), (15) and the successive terms (13), (14) are the power series expansion of the integral equation K + ^(2,l) = K + (2,l) -i(K + {2, 3)A(3)K + < a \3, l)dr 3 , (16) which it also satisfies. We would now expect to choose, for the special solu- tion of (12), K+ = K where #(,(2, 1) vanishes for t 2 <ti and for t 2 >h is given by (3) where #„ and E n are the eigenfunctions and energy values of a particle satis- fying Dirac's equation, and </>„* is replaced by #„. The formulas arising from this choice, however, suffer from the drawback that they apply to the one electron theory of Dirac rather than to the hole theory of the positron. For example, consider as in Fig. 1(a) an electron after being scattered by a potential in a small region 3 of space time. The one electron theory says (as does (3) with K + = K ) that the scattered amplitude at another point 2 will proceed toward positive times with both positive and negative energies, that is with both positive and negative rates of change of phase. No wave is scattered to times previous to the time of scattering. These are just the properties of K (2, 3). On the other hand, according to the positron theory negative energy states are not available to the electron after the scattering. Therefore the choice K + = Ko is unsatisfactory. But there are other solutions of (12). We shall choose the solution defining K+(2, 1) so that K + {2, 1) for h>h is the sum of (3) over positive energy states only. Now this new solution must satisfy (12) for all times in order that the representation be complete. It must therefore differ from the old solution K by a solution of the homogeneous Dirac equation. It is clear from the definition that the difference K — K + is the sum of (3) over all negative energy states, as long as t 2 >h. But this difference must be a solution of the homogeneous Dirac equation for all times and must therefore be represented by the same sum over negative energy states also for t 2 <h. Since ^0 = in this case, it follows that our new kernel, K+(2, l),for t 2 <h is the negative of the sum (J) over negative energy states. That is, K + (2,l) = 2Zpos En <l>n(2)tn(l) Xexp(-iE n (t 2 -h)) for t 2 >h (17) -2ZNEGE n <t>n(2)$ n (l) Xexp(-iE n (l 2 -ti)) for t 2 <h With this choice of K + our equations such as (13) and (14) will now give results equivalent to those of the positron hole theory. That (14), for example, is the correct second order expression for finding at 2 an electron originally at 1 according to the positron theory may be seen as follows (Fig. 2). Assume as a special example that t 2 >h and that the potential vanishes except in interval t 2 —ti so that U and / 3 both lie between /i and t 2 . First suppose l 4 >t 3 (Fig. 2(b)). Then (since / 3 >/i) 171 THEORY OF POSITRONS 753 the electron assumed originally in a positive energy state propagates in that state (by K+(3, 1)) to position 3 where it gets scattered (.4(3)). It then proceeds to 4, which it must do as a positive energy electron. This is correctly described by (14) for j£+(4, 3) contains only positive energy components in its expansion, as ti>h. After being scattered at 4 it then proceeds on to 2, again necessarily in a positive energy state, as t 2 >U. In positron theory there is an additional contribution due to the possibility of virtual pair production (Fig. 2(c)). A pair could be created by the potential 4(4) at 4, the electron of which is that found later at 2. The positron (or rather, the hole) proceeds to 3 where it annihilates the electron which has arrived there from 1. This alternative is already included in (14) as con- tributions for which ti<h, and its study will lead us to an interpretation of K + (A, 3) for ti<h. The factor K + (2, 4) describes the electron (after the pair produc- tion at 4) proceeding from 4 to 2. Likewise K + (3, 1) represents the electron proceeding from 1 to 3. K + (A, 3) must therefore represent the propagation of the positron or hole from 4 to 3. That it does so is clear. The fact that in hole theory the hole proceeds in the manner of and electron of negative energy is reflected in the fact that K+(4, 3) for h<h is (minus) the sum of only negative energy components. In hole theory the real energy of these intermediate states is, of course, positive. This is true here too, since in the phases exp(— iE n {h— h)) defining K+(4, 3) in (17), En is nega- tive but so is t i — h. That is, the contributions vary with t 3 as exp(— i\E n \(h— h)) as they would if the energy of the intermediate state were \E n \. The fact that the entire sum is taken as negative in computing if+(4, 3) is reflected in the fact that in hole theory the amplitude has its sign reversed in accordance with the Pauli principle and the fact that the electron arriving at 2 has been exchanged with one in the sea. 6 To this, and to higher orders, all processes involving virtual pairs are correctly described in this way. The expressions such as (14) can still be described as a passage of the electron from 1 to 3 (K+(3, 1)), scatter- ing at 3 by 4(3), proceeding to 4 (i^+(4, 3)), scattering again, 4(4), arriving finally at 2. The scatterings may, however, be toward both future and past times, an electron propagating backwards in time being recog- nized as a positron. This therefore suggests that negative energy com- ponents created by scattering in a potential be con- sidered as waves propagating from the scattering point toward the past, and that such waves represent the propagation of a positron annihilating the electron in the potential. 7 6 It has often been noted that the one-electron theory apparently gives the same matrix elements for this process as does hole theory. The problem is one of interpretation, especially in a way that will also give correct results for other processes, e.g., self-energy. 7 The idea that positrons can be represented as electrons with proper time reversed relative to true time has been discussed by the author and others, particularly by Stuckelberg. E. C. C. With this interpretation real pair production is also described correctly (see Fig. 3). For example in (13) if h<h<h the equation gives the amplitude that if at time /i one electron is present at 1, then at time / 2 just one electron will be present (having been scattered at 3) and it will be at 2. On the other hand if h is less than h, for example, if h—t\<tz, the same expression gives the amplitude that a pair, electron at 1, positron at 2 will annihilate at 3, and subsequently no particles will be present. Likewise if h and t x exceed k we have (minus) the amplitude for finding a single pair, electron at 2, positron at 1 created by 4(3) from a vacuum. If h>h>h, (13) describes the scattering of a positron. All these amplitudes are relative to the amplitude that a vacuum will remain a vacuum, which is taken as unity. (This will be discussed more fully later.) The analogue of (2) can be easily worked out. 8 It is, *(2) J- (2, \)N{\)*(\)dW (18) where dWi is the volume element of the closed 3- dimensional surface of a region of space time containing Fig. 3. Several different processes can be described by the same formula depending on the time relations of the variables tz, h. Thus P v \K + ^{2, 1)| 2 is the probability that: (a) An electron at 1 will be scattered at 2 (and no other pairs form in vacuum). (b) Electron at 1 and positron at 2 annihilate leaving nothing. (c) A single pair at 1 and 2 is created from vacuum, (d) A positron at 2 is scattered to 1. (K+. u >(2, 1) is the sum of the effects of scattering in the potential to all orders. P v is a normalizing constant.) Stuckelberg, Helv. Phys. Acta 15, 23 (1942); R. P. Feynman, Phys. Rev. 74, 939 (1948). The fact that classically the action (proper time) increases continuously as one follows a trajectory is reflected in quantum mechanics in the fact that the phase, which is \E n \ \h—li\, always increases as the particle proceeds from one scattering point to the next. 8 By multiplying (12) on the right by (— tVi— m) and noting that Vi«(2, 1) = -Vj5(2, 1) show that .£+(2,1) also satisfies K+(2, 1)(— iV\— w)=z'5(2, 1), where the Vi operates on variable 1 in K+(2, 1) but is written after that function to keep the correct order of the y matrices. Multiply this equation by ^(1) and Eq. (11) (with 4 = 0, calling the variables 1) by K+(2, 1), subtract and integrate over a region of space-time. The integral on the left- hand side can be transformed to an integral over the surface of the region. The right-hand side is \p(2) if the point 2 lies within the region, and is zero otherwise. (What happens when the 3- surface contains a light line and hence has no unique normal need not concern us as these points can be made to occur so far away from 2 that their contribution vanishes.) 172 754 FE YN M AN point 2, and N(l) is ^(1)7,, where A^ M (1) is the inward drawn unit normal to the surface at the point 1. That is, the wave function ^(2) (in this case for a free par- ticle) is determined at any point inside a four-dimen- sional region if its values on the surface of that region are specified. To interpret this, consider the case that the 3-surface consists essentially of all space at some time say t = previous to t 2 , and of all space at the time T>t 2 . The cylinder connecting these to complete the closure of the surface may be very distant from x 2 so that it gives no appreciable contribution (as K + (2, 1) decreases expo- nentially in space-like directions). Hence, if 74 = /?, since the inward drawn normals TV" will be /? and — /3, *(2): jV (2, i)W{i)ffixi / K + (2, VW(l')(Pxv, (19) where /i = 0, ty — T. Only positive energy (electron) components in ^(1) contribute to the first integral and only negative energy (positron) components of ^(1') to the second. That is, the amplitude for finding a charge at 2 is determined both by the amplitude for finding an electron previous to the measurement and by the amplitude for finding a positron after the measurement. This might be interpreted as meaning that even in a problem involving but one charge the amplitude for finding the charge at 2 is not determined when the only thing known in the amplitude for finding an electron (or a positron) at an earlier time. There may have been no electron present initially but a pair was created in the measurement (or also by other external fields). The amplitude for this contingency is specified by the amplitude for finding a positron in the future. We can also obtain expressions for transition ampli- tudes, like (5). For example if at / = we have an elec- tron present in a state with (positive energy) wave function /(x), what is the amplitude for finding it at t=T with the (positive energy) wave function g(x)? The amplitude for finding the electron anywhere after / = is given by (19) with ^(1) replaced by /(x), the second integral vanishing. Hence, the transition ele- ment to find it in state g(x) is, in analogy to (5), just (t 2 =T,t 1 = 0) I g{x 2 )&K + {2, DjS/CxOAidhc, (20) since g* = <?/3. If a potential acts somewhere in the interval between and T, K+ is replaced by K+ (A) . Thus the first order effect on the transition amplitude is, from (13), -if g(x 2 )0K + (2, 3)A(3)K + (3, l)/3/(x 1 )rf 3 x 1 d 3 x 2 . (21) Expressions such as this can be simplified and the 3-surface integrals, which are inconvenient for rela- tivistic calculations, can be removed as follows. Instead of defining a state by the wave function /(x), which it has at a given time /i = 0, we define the state by the function /(l) of four variables Xi, h which is a solution of the free particle equation for all h and is /(xi) for /i = 0. The final state is likewise defined by a function g(2) over-all space-time. Then our surface integrals can be performed since fK + (3, l) i 8/(xi)d 3 x 1 =/(3) and fg(x 2 )P(Px 2 K + (2, 3) = 0(3). There results ifg(3)A(3)f(3)di (22) the integral now being over-all space-time. The transi- tion amplitude to second order (from (14)) is -IS g(2)A(2)K + (2, l)A{\)f{\)dT,dr 2 , (23) for the particle arriving at 1 with amplitude /(l) is scattered (.4(1)), progresses to 2, (K + (2, 1)), and is scattered again (.4(2)), and we then ask for the ampli- tude that it is in state g(2). If g(2) is a negative energy state we are solving a problem of annihilation of elec- tron in /(l), positron in g(2), etc. We have been emphasizing scattering problems, but obviously the motion in a fixed potential V, say in a hydrogen atom, can also be dealt with. If it is first viewed as a scattering problem we can ask for the amplitude, 0*(1), that an electron with original free wave function was scattered k times in the potential V either forward or backward in time to arrive at 1. Then the amplitude after one more scattering is **+i(2) = -iJK42, DVmMVdn. (24) An equation for the total amplitude *(i)«£ **(i) ft=0 for arriving at 1 either directly or after any number of scatterings is obtained by summing (24) over all k from Oto 00; lK2) = o (2) -•/* (2,1)7(1)^(1)^7 (25) Viewed as a steady state problem we may wish, for example, to find that initial condition 4> (or better just the \p) which leads to a periodic motion of \f/. This is most practically done, of course, by solving the Dirac equation, (tV-»)*(l)«K(l)*(l), (26) deduced from (25) by operating on both sides by iV 2 —m, thereby eliminating the </> , and using (12). This illus- trates the relation between the points of view. For many problems the total potential A-\- V may be split conveniently into a fixed one, V, and another, A, considered as a perturbation. If A' + (n is defined as in 173 THEORY OF POSITRONS 755 (16) with V for A, expressions such as (23) are valid and useful with K+ replaced by K+ (V) and the functions /(l), g(2) replaced by solutions for all space and time of the Dirac Eq. (26) in the potential V (rather than free particle wave functions). 4. PROBLEMS INVOLVING SEVERAL CHARGES We wish next to consider the case that there are two (or more) distinct charges (in addition to pairs they may produce in virtual states). In a succeeding paper we discuss the interaction between such charges. Here we assume that they do not interact. In this case each particle behaves independently of the other. We can expect that if we have two particles a and b, the ampli- tude that particle a goes from Xi at h, to x 3 at t 3 while b goes from x 2 at t% to x 4 at U is the product K{3, 4; 1, 2) = K +a (3, l)K +b (4, 2). The symbols a, b simply indicate that the matrices appearing in the K + apply to the Dirac four component spinors corresponding to particle a or b respectively (the wave function now having 16 indices). In a potential K +a and K +b become K + J A) and K +b ^ where K +a ^ A > is defined and calculated as for a single particle. They commute. Hereafter the a, b can be omitted; the space time variable appearing in the kernels suffice to define on what they operate. The particles are identical however and satisfy the exclusion principle. The principle requires only that one calculate K{3, 4; 1, 2)-X(4, 3; 1, 2) to get the net amplitude for arrival of charges at 3, 4. (It is normalized assuming that when an integral is performed over points 3 and 4, for example, since the electrons represented are identical, one divides by 2.) This expression is correct for positrons also (Fig. 4). For example the amplitude that an electron and a positron found initially at x x and x 4 (say ti = ti) are later found at x 3 and x 2 (with t2 = tz>h) is given by the same expression K + ^(3, l)K + ^(4, 2)-# + ">(4, l)K + ^(3, 2). (27) The first term represents the amplitude that the electron proceeds from 1 to 3 and the positron from 4 to 2 (Fig. 4(c)), while the second term represents the interfering amplitude that the pair at 1, 4 annihilate and what is found at 3, 2 is a pair newly created in the potential. The generalization to several particles is clear. There is an additional factor K + U) for each particle, and anti- symmetric combinations are always taken. No account need be taken of the exclusion principle in intermediate states. As an example consider again expression (14) for h>h and suppose U<h so that the situation represented (Fig. 2(c)) is that a pair is made at 4 with the electron proceeding to 2, and the positron to 3 where it annihilates the electron arriving from 1. It may be objected that if it happens that the electron created at 4 is in the same state as the one coming from 1, then the process cannot occur because of the exclusion principle and we should not have included it in our (c) Fig. 4. Some problems involving two distinct charges (in addi- tion to virtual pairs they may produce) : P v \K + ( - A, (3, \)K + (A) (4, 2) -#+ u >(4, 1)JC + M)(3, 2) | 2 is the probability that: (a) Electrons at 1 and 2 are scattered to 3, 4 (and no pairs are formed), (b) Starting with an electron at 1 a single pair is formed, positron at 2, electrons at 3, 4. (c) A pair at 1, 4 is found at 3, 2, etc. The exclu- sion principle requires that the amplitudes for processes involving exchange of two electrons be subtracted. term (14). We shall see, however, that considering the exclusion principle also requires another change which reinstates the quantity. For we are computing amplitudes relative to the amplitude that a vacuum at h will still be a vacuum at t 2 . We are interested in the alteration in this amplitude due to the presence of an electron at 1. Now one process that can be visualized as occurring in the vacuum is the creation of a pair at 4 followed by a re-annihilation of the same pair at 3 (a process which we shall call a closed loop path). But if a real electron is present in a certain state 1, those pairs for which the electron was created in state 1 in the vacuum must now be excluded. We must therefore subtract from our relative amplitude the term corresponding to this process. But this just rein- states the quantity which it was argued should not have been included in (14), the necessary minus sign coming automatically from the definition of K + . It is obviously simpler to disregard the exclusion principle completely in the intermediate states. All the amplitudes are relative and their squares give the relative probabilities of the various phenomena. Absolute probabilities result if one multiplies each of the probabilities by P v , the true probability that if one has no particles present initially there will be none finally. This quantity P v can be calculated by normal- izing the relative probabilities such that the sum of the probabilities of all mutually exclusive alternatives is unity. (For example if one starts with a vacuum one can calculate the relative probability that there remains a 174 756 R. P. FE YNM AN vacuum (unity), or one pair is created, or two pairs, etc. The sum is P„ -1 .) Put in this form the theory is com- plete and there are no divergence problems. Real proc- esses are completely independent of what goes on in the vacuum. When we come, in the succeeding paper, to deal with interactions between charges, however, the situation is not so simple. There is the possibility that virtual elec- trons in the vacuum may interact electromagnetically with the real electrons. For that reason processes occur- ing in the vacuum are analyzed in the next section, in which an independent method of obtaining P v is discussed. 5. VACUUM PROBLEMS An alternative way of obtaining absolute amplitudes is to multiply all amplitudes by C v , the vacuum to vacuum amplitude, that is, the absolute amplitude that there be no particles both initially and finally. We can assume C v = 1 if no potential is present during the interval, and otherwise we compute it as follows. It differs from unity because, for example, a pair could be created which eventually annihilates itself again. Such a path would appear as a closed loop on a space-time diagram. The sum of the amplitudes resulting from all such single closed loops we call L. To a first approxima- tion L is L«: -ffsKK + (2,l)A(X) XK + (l,2)A(2)yT l dT2. (28) For a pair could be created say at 1, the electron and positron could both go on to 2 and there annihilate. The spur, Sp, is taken since one has to sum over all possible spins for the pair. The factor \ arises from the fact that the same loop could be considered as starting at either potential, and the minus sign results since the interactors are each —iA. The next order term would be 9 D» = + (i/3)fffsptK + (2, 1)4(1) XK+(1, 3)A(3)K+(3, 2)i4(2)]dT 1 dT 2 rfT 3 , etc. The sum of all such terms gives Z,. 10 9 This term actually vanishes as can be seen as follows. In any spur the sign of all y matrices may be reversed. Reversing the sign of 7 in K+(2, 1) changes it to the transpose of K + (l, 2) so that the order of all factors and variables is reversed. Since the integral is taken over all n, r 2 , and t% this has no effect and we are left with (— l) 3 from changing the sign of A. Thus the spur equals its negative. Loops with an odd number of potential interactors give zero. Physically this is because for each loop the electron can go around one way or in the opposite direction and we must add these amplitudes. But reversing the motion of an electron makes it behave like a positive charge thus changing the sign of each potential interaction, so that the sum is zero if the number of interactions is odd. This theorem is due to W. H. Furry, Phys. Rev. 51, 125(1937). 10 A closed expression for L in terms of K+<- A) is hard to obtain because of the factor (1/n) in the nth term. However, the per- turbation in L, AL due to a small change in potential AA, is easy to express. The (l/«) is canceled by the fact that AA can appear In addition to these single loops we have the possi- bility that two independent pairs may be created and each pair may annihilate itself again. That is, there may be formed in the vacuum two closed loops, and the contribution in amplitude from this alternative is just the product of the contribution from each of the loops considered singly. The total contribution from all such pairs of loops (it is still consistent to disregard the exclusion principle for these virtual states) is L 2 /2 for in L 2 we count every pair of loops twice. The total vacuum-vacuum amplitude is then C v =l-L+L 2 /2-L*/6+ ixp(-L), (30) the successive terms representing the amplitude from zero, one, two, etc., loops. The fact that the contribu- tion to C v of single loops is — L is a consequence of the Pauli principle. For example, consider a situation in which two pairs of particles are created. Then these pairs later destroy themselves so that we have two loops. The electrons could, at a given time, be inter- changed forming a kind of figure eight which is a single loop. The fact that the interchange must change the sign of the contribution requires that the terms in C v appear with alternate signs. (The exclusion principle is also responsible in a similar way for the fact that the amplitude for a pair creation is —K+ rather than +K + .) Symmetrical statistics would lead to C„ = 1 + L+ L 2 /2 = exp(+ L) . The quantity L has an infinite imaginary part (from L m , higher orders are finite). We will discuss this in connection with vacuum polarization in the succeeding paper. This has no effect on the normalization constant for the probability that a vacuum remain vacuum is given by Pv= |C„| 2 =exp(-2-real part of L), from (30). This value agrees with the one calculated directly by renormalizing probabilities. The real part of L appears to be positive as a consequence of the Dirac equation and properties of K + so that P v is less than one. Bose statistics gives C = exp(+-£) and conse- quently a value of P v greater than unity which appears meaningless if the quantities are interpreted as we have done here. Our choice of K+ apparently requires the exclusion principle. Charges obeying the Klein-Gordon equation can be equally well treated by the methods which are dis- cussed here for the Dirac electrons. How this is done is discussed in more detail in the succeeding paper. The real part of L comes out negative for this equation so that in this case Bose statistics appear to be required for consistency. 3 in any of the n potentials. The result after summing over n by (13), (14) and using (16) is AL=-ifspl(K^Ki, i)~K + (l, l))Ail(l)]rfTi. (29) The term #+(1, 1) actually integrates to zero. 175 THEORY OF POSITRONS 757 6. ENERGY-MOMENTUM REPRESENTATION The practical evaluation of the matrix elements in some problems is often simplified by working with momentum and energy variables rather than space and time. This is because the function K + (2, 1) is fairly complicated but we shall find that its Fourier transform is very simple, namely (i/ATr 2 ){p—m)- x that is K + (2,l) = (i/W) /<>- i)- 1 exp(-ip-x 2 i)d i p, (31) where p-x 2 i=p-x 2 —p-xi=p )l x 2 ^—p ll xi ll , P=p v n ll ., and d*p means (2ir)~ 2 dpidp 2 dpzdpi, the integral over all p. That this is true can be seen immediately from (12), for the representation of the operator iV— m in energy (pi) and momentum (pi, 2, 3) space is p— m and the trans- form of 5(2, 1) is a constant. The reciprocal matrix (p—m)- 1 can be interpreted as (/i+ffl)^-^)" 1 for pp—m 2 = {p—m)(p-\-m) is a pure number not involving 7 matrices. Hence if one wishes one can write where K + (2,l) = i(iV 2 +m)I + (2,l), I + (2, 1) = (2tt)- 2 f(^-»V exp(-ip-x n )d% (32) is not a matrix operator but a function satisfying □ 2 2 / + (2, \)-m 2 I + (2, 1) = 8(2, 1), (33) where -[J 2 2 = (V 2 ) 2 = (d/dx u )(d/dx 2lt ). The integrals (31) and (32) are not yet completely defined for there are poles in the integrand when pp—m 2 =0. We can define how these poles are to be evaluated by the rule that m is considered to have an infinitesimal negative imaginary part. That is m, is re- placed by m— id and the limit taken as 5— >0 from above. This can be seen by imagining that we calculate K+ by integrating on pi first. If we call E= + (m?+pi 2 +/ > 2 2 +/>3 2 ) i then the integrals involve pi essentially as S exp(— ipi(t 2 — h))dpi(pi 2 — E?)~ l which has poles at p t =+E and pi=—E. The replacement of m by m— i8 means that E has a small negative imaginary part; the first pole is below, the second above the real axis. Now if t 2 — h>0 the contour can be completed around the semicircle below the real axis thus giving a residue from the pi=+E pole, or -(2E)~ l exp(-iE(t 2 -ti)). If k— h<0 the upper semicircle must be used, and Pi=—Ea.t the pole, so that the function varies in each case as required by the other definition (17). Other solutions of (12) result from other prescrip- tions. For example if pi in the factor (p 2 —m 2 )~ l is con- sidered to have a positive imaginary part K+ becomes replaced by K , the Dirac one-electron kernel, zero for t 2 <t\. Explicitly the function is 11 (x, t=x 2 i li ) I + (x, t)=- (iir)- 1 8(s 2 )+ (m/Zirs)!!^ (ms), (34) where s= + (t 2 -x 2 )* for fi>x 2 and s= -i(x 2 -t 2 )* for n I+{x,t) is {2i)-i{Di{x,t)-iD{x,t)) where D, and D are the functions defined by W. Pauli, Rev. Mod. Phys. 13, 203 (1941). t 2 <x 2 , Hi (2) is the Hankel function and 8(s 2 ) is the Dirac delta function of s 2 . It behaves asymptotically as exp(— ims), decaying exponentially in space-like directions. 12 By means of such transforms the matrix elements like (22), (23) are easily worked out. A free particle wave function for an electron of momentum pi is U\ exp(— ipi-x) where u\ is a constant spinor satisfying the Dirac equation p\U\ = mu\ so that pi 2 = m 2 . The matrix element (22) for going from a state p u u x to a state of momentum p 2 , spinor u%, is — Air 2 i(u 2 a(q)ui) where we have imagined A expanded in a Fourier integral A(\) = I a(q) exp(—iq-xi)d 4 q, and we select the component of momentum q=p 2 —p\. The second order term (23) is the matrix element between U\ and u 2 of -Wi((a(p 2 ■pi-q))(J>i+q-m)- l a(q)d*q, (35) since the electron of momentum pi may pick up q from the potential a(q), propagate with momentum pi+q (factor (p\-\-q— w) _1 ) until it is scattered again by the potential, a(p 2 —pi—q), picking up the remaining mo- mentum, p 2 —pi—q, to bring the total to p 2 . Since all values of q are possible, one integrates over q. These same matrices apply directly to positron prob- lems, for if the time component of, say, pi is negative the state represents a positron of four-momentum —pi, and we are describing pair production if p 2 is an elec- tron, i.e., has positive time component, etc. The probability of an event whose matrix element is (u 2 Mui) is proportional to the absolute square. This may also be written (uiMu 2 )(u 2 Mui), where M is M with the operators written in opposite order and explicit appearance of i changed to — i(M is /3 times the complex conjugate transpose of /3M). For many problems we are not concerned about the spin of the final state. Then we can sum the probability over the two u 2 corresponding to the two spin directions. This is not a complete set be- cause p 2 has another eigenvalue, — m. To permit sum- ming over all states we can insert the projection operator (2m)~ l (p 2 -\-m) and so obtain (2m)~ l (iiiM(p 2 -\-m)Mui) for the probability of transition from pi, u u to p 2 with arbitrary spin. If the incident state is unpolarized we can sum on its spins too, and obtain (2 m)~ 2 Sp\_(pi+ m)M(p 2 + m)M ] (36) for (twice) the probability that an electron of arbitrary spin with momentum pi will make transition to p 2 . The expressions are all valid for positrons when p's with 12 If the —id is kept with m here too the function I + approaches zero for infinite positive and negative times. This may be useful in general analyses in avoiding complications from infinitely remote surfaces. 176 758 FEYNMAN negative energies are inserted, and the situation inter- preted in accordance with the timing relations discussed above. (We have used functions normalized to (uu) = 1 instead of the conventional (uj3u) = (u*u) = 1 . On our scale (ufiu) = energy/ w so the probabilities must be corrected by the appropriate factors.) The author has many people to thank for fruitful conversations about this subject, particularly H. A. Bethe and F. J. Dyson. APPENDIX a. Deduction from Second Quantization In this section we shall show the equivalence of this theory with the hole theory of the positron. 2 According to the theory of second quantization of the electron field in a given potential, 13 the state of this field at any time is represented by a wave function x satisfying idx/dt=H X , where H= y**(x)(a-(-iV-A)+^ 4 +w/3)*(x)(f 3 x and *(x) is an operator annihilating an electron at position x, while **(x) is the corresponding creation operator. We contemplate a situation in which at t = we have present some electrons in states repre- sented by ordinary spinor functions /i(x), /2(x), • ■ ■ assumed orthogonal, and some positrons. These are described as holes in the negative energy sea, the electrons which would normally fill the holes having wave functions pi(x'), pi{x), •••. We ask, at time T what is the amplitude that we find electrons in states gi(x), gt(x), ■ ■ ■ and holes at qi(x),qz(x), • • •. If the initial and final state vectors representing this situation are xi and xi respectively, we wish to calculate the matrix element R = \xi* exp(-^ r #^)xi)=(x/*Sx.). (37) We assume that the potential A differs from zero only for times between and T so that a vacuum can be defined at these times. If xo represents the vacuum state (that is, all negative energy states filled, all positive energies empty), the amplitude for having a vacuum at time T, if we had one at / = 0, is C^=(xo*5xo), (38) writing 5 for exp( — if T Hdt). Our problem is to evaluate R and show that it is a simple factor times C„, and that the factor involves the K + iA > functions in the way discussed in the previous sections. To do this we first express x; in terms of xo- The operator / **(x)0(x)rf 3 x, (39) creates an electron with wave function <j>(x). Likewise <f>= f<t>*(x) X*(x)<f 3 x annihilates one with wave function 4>{x). Hence state x. is xi = Fi*F?*- ■ PiP-2- • -xo while the final state is 6VG 2 *- ' " XQ\Qi- ■ -xo where F,-, G„ Pi, Qi are operators defined like <I>, in (39), but with/, gi, pi, qi replacing <)>; for the initial state would result from the vacuum if we created the electrons in j\, / 2 , and annihilated those in pi, pi, ••-. Hence we must find R = (xo* • • ■ QSQi* ■ ■ ■ G i G 1 SF 1 *F i * • • • P,P 2 • • • xo) . (40) To simplify this we shall have to use commutation relations be- tween a <i>* operator and S. To this end consider exp( — iJl l Hdt')$* Xexp(+iJl'Hdt') and expand this quantity in terms of **(x), giving f**(x)<t>(x,t)d 3 x, (which defines <t>(x,t)). Now multiply this equation by exp(+iJl'Hdt') ■ ■ -exp(-iJl'Hdt') and find JV(x)tf>(x)<f 3 x = JV(x, l)4>(x, t)d% (41) where we have defined ¥(x, t) by V(x, l) = exp(+if 'Hdt')y(x) 13 See, for example, G. Wentzel, Einfuhrung in die Quanten- theorie der Wellenfeldcr (Franz Deuticke, Leipzig, 1943), Chap- ter V. Xexpi — iJl'Hdt'). As is well known *(x, t) satisfies the Dirac equation, (differentiate *(x, t) with respect to t and use commuta- tion relations of H and *) id¥(x, 0/d<=(a-(-*V-A)-Md-*»/3)*(x, I). (42) Consequently </>(x, t) must also satisfy, the Dirac equation (differ- entiate (41) with respect to t, use (42) and integrate by parts). That is, if <t>(x, T) is that solution of the Dirac equation at time T which is <t>(x) at t=0, and if we define ** = y**(x)^(x)d 3 x and *'* = y**(x)<>(x, T)d 3 x then *'* = 5**5 _1 , or S** = <*>'*£. (43) The principle on which the proof will be based can now be illustrated by a simple example. Suppose we have just one electron initially and finally and ask for ■■(xo*GSF*xo)- (44) We might try putting F* through the operator 5 using (43), SF* = F'*S, where/' in F'* : = y**(x)/'(x)<f 3 x is the wave function at T arising from /(x) at 0. Then r=(xo*GF'*Sx a )=fg*(x)f'(x)d*x-C„-(xo*F'*GS X o), (45) where the second expression has been obtained by use of the defi- nition (38) of C v and the general commutation relation GF*+F*G = fg*(x)f(x)d% which is a consequence of the properties of *(x) (the others are FG= —GF and F*G*=—G*F*). Now xo*F'* in the last term in (45) is the complex conjugate of F'xo- Thus if/' contained only positive energy components, f'xo would vanish and we would have reduced r to a factor times C v . But F', as worked out here, does contain negative energy components created in the potential A and the method must be slightly modified. Before putting F* through the operator we shall add to it another operator F"* arising from a function f"(x) containing only negative energy components and so chosen that the resulting /' has only positive ones. That is we want S(F poe *+F aeg "*) = F poa '*S, (46) where the "pos" and "neg" serve as reminders of the sign of the energy components contained in the operators. This we can now use in the form SF poB * = F poa '*S-SFne g "*. (47) In our one electron problem this substitution replaces r by two terms ,= (xo*C7F po8 '*Sxo)-(xo*GSF The first of these reduces to *xo). ' = JV«/poa'(x)d 3 X-C„ as above, for Fp OS 'xo is now zero, while the second is zero since the creation operator F aes "* gives zero when acting on the vacuum state as all negative energies are full. This is the central idea of the demonstration. The problem presented by (46) is this: Given a function / pos (x) at time 0, to find the amount, /neg", of negative energy component which must be added in order that the solution of Dirac's equa- tion at time T will have only positive energy components, / po3 '. This is a boundary value problem for which the kernel K + U) is designed. We know the positive energy components initially, / pos , and the negative ones finally (zero). The positive ones finally are therefore (using (19)) /po.'(x*)=/jr + W>(2, l)/3/ po9 (x 1 )d 3 x 1 , (48) where U_= T, t x = Q. Similarly, the negative ones initially are /neg"(x 2 ) =/aV- 4) (2, l)/3/ P os(Xl)<f 3 X 1 -/ P os(x 2 ), (49) where h approaches zero from above, and ii = 0. The / pos (x>) is 177 THEORY OF POSITRONS 759 subtracted to keep in / ne g"(x 2 ) only those waves which return from the potential and not those arriving directly at t 2 from the K + (2, 1) part of AV X) (2, 1), as lz— -0. We could also have written The value of Cv(k— At ) arises from the Hamiltonian II to— Ma which differs from Hi just by having an extra potential during the short interval A/ - Hence, to first order in Al , we have Therefore the one-electron problem gives by (48) r = fg*(x)f poa '(x)d 3 x-C„, /«.«"<**) =/lX+ W) (2, l)-A\(2,l)]/3/ po8 (x I )^x 1 . (50) CMo-Uo) = (xo*exp(-if t T _ M 7/« -a^)xo) = (xo* exp(- if* Htodt)\l-iM f**(x) X(-a-A(x,<o)+^ 4 (x,/ ))*(x^ 3 xJxo); we therefore obtain for the derivative of Cv the expression -dC c (lo)/dt = -i(xo* exp(-if T HtodtJ xJV(x)/3/l(x,/o)*(x)rf 3 xxo), (51) ■■ Cvfg*(x 2 )K+^(2, VM*i)d 3 Xid 3 x 2 , as expected in accordance with the reasoning of the previous sec- tions (i.e., (20) with AV X) replacing A' + ). The proof is readily extended to the more general expression R, (40), which can be analyzed by induction. First one replaces Pi* by a relation such as (47) obtaining two terms R=(xo*- ■ -Qi*Qi*- ■ -GzG^po^SF,*-- -P X P 2 - • - X o) - (xo* • • ■ Q2*Qi* ■ ■ ■ G 2 G l SF laeg "*F 2 * ■ ■ ■ PJ 2 • • • xo) ■ In the first term the order of Pi pos '* and Gi is then interchanged, producing an additional term ygi*(x)/i P0 s'(x)<f 3 x times an expres- sion with one less electron in initial and final state. Next it is exchanged with G 2 producing an addition — J % g 2 *(x)fi poa '(x)d i x times a similar term, etc. Finally on reaching the Q* with which it anticommutes it can be simply moved over to juxtaposition with xo* where it gives zero. The second term is similarly handled by moving Fmeg"* through anti commuting F 2 *, etc., until it reaches Pi. Then it is exchanged with P x to produce an addi- tional simpler term with a factor = Fj , pi*(x)fi ueli "(x)d 3 x or z ¥fpi*(x2)K+^(2, lWKxO^XKPxjfrom (49), with ^ = ^ = (the extra /i(x 2 ) in (49) gives zero as it is orthogonal to pi(x 2 )). This describes in the expected manner the annihilation of the pair, electron /i, positron pi. The F aeg "* is moved in this way succes- sively through the P's until it gives zero when acting on xo- Thus R is reduced, with the expected factors (and with alternating signs as required by the exclusion principle), to simpler terms containing two less operators which may in turn be further reduced by using P 2 * in a similar manner, etc. After all the P* are used the Q*'s can be reduced in a similar manner. They are moved through the 5 in the opposite direction in such a manner as to produce a purely negative energy operator at time 0, using relations analogous to (46) to (49). After all this is done we are left simply with the ex- pected factor times C» (assuming the net charge is the same in initial and final state.) In this way we have written the solution to the general problem of the motion of electrons in given potentials. The factor Cv is obtained by normalization. However for photon fields it is desir- able to have an explicit form for Cv in terms of the potentials. This is given by (30) and (29) and it is readily demonstrated that this also is correct according to second quantization. b. Analysis of the Vacuum Problem We shall calculate Cv from second quantization by induction considering a series of problems each containing a potential dis- tribution more nearly like the one we wish. Suppose we know C v for a problem like the one we want and having the same potentials for time t between some to and T, but having potential zero for times from to to. Call this Cvito), the corresponding Hamiltonian Hto and the sum of contributions for all single loops, L(t a ). Then for to= T we have zero potential at all times, no pairs can be produced, L(T) = and Cv(T) = l. For t a =0 we have the com- plete problem, so that C v (0) is what is defined as Cv in (38). Generally we have, :*('o) = (xo* exp(-^ T Htodtjy = {xo*expy-if t Htodtjxoj, since Hto is identical to the constant vacuum Hamiltonian Ht for Kto and xo is an eigenfunction of Ht with an eigenvalue (energy of vacuum) which we can take as zero. which will be reduced to a simple factor times Cv{t a ) by methods analogous to those used in reducing R. The operator * can be imagined to be split into two pieces ^pos and ^oeg operating on positive and negative energy states respectively. The * pos on xo gives zero so we are left with two terms in the current density, ■tpos^^neg and ¥ ae g*0Ay Dee . The latter ^neg^^vE'neg is just •the expectation value of (5A taken over all negative energy states (minus ^neg/^^neg* which gives zero acting on xo)- This is the effect of the vacuum expectation current of the electrons in the sea which we should have subtracted from our original Hamil- tonian in the customary way. The remaining term ^ p03 */3^4^neg, or its equivalent SE f pos */3i4*' can be considered as **(x)f p03 (x) where f pos (x) is written for the positive energy component of the operator QA^ (x). Now this operator, *'*(x)f pos (x), or more precisely just the **(x) part of it, can be pushed through the exp( — iJlo T Hdl) in a manner exactly analogous to (47) when / is a function. (An alternative derivation results from the consideration that the operator ^(x, t) which satisfies the Dirac equation also satisfies the linear integral equa- tions which are equivalent to it.) That is, (51) can be written by (48), (50), -dCv(to)/dt = -i(xo*ff**(x2)K + ^(2, 1) Xexp(-*J^ fftf^aMxO^x^Xaxo) + *(xo* exp(-ij^ Hdt)ff**(x 2 )lK + ^(2, 1) -K + (2, l)2A(l)^(x 1 )d 3 x 1 d 3 x 2X o) i where in the first term t 2 =T, and in the second t 2 -*t =ti. The (A) in K + (A) refers to that part of the potential A after t . The first term vanishes for it involves (from the K + u) (2, 1)) only positive energy components of **, which give zero operating into xo*. In the second term only negative components of ^*(x 2 ) appear. If, then ^*(x 2 ) is interchanged in order with ^(xi) it will give zero operating on xo, and only the term, -dC,(to)/dh=+ifspl(K + w(l, 1) -K+a, l)M(l)]i 3 xi-C„(/ ), (52) will remain, from the usual commutation relation of ^* and ^. The factor of Cv(t ) in (52) times -Mo is, according to (29) (reference 10), just L(t — At ) — L(t ) since this difference arises from the extra potential AA = A during the short time interval A* . Hence -dCv(to)/dt = + (dL(to)/dto)Cv(t ) so that integration from h= T to t =0 establishes (30). Starting from the theory of the electromagnetic field in second quantization, a deduction of the equations for quantum electro- dynamics which appear in tne succeeding paper may be worked out using very similar principles. The Pauli-Weisskopf theory of the Klein-Gordon equation can apparently be analyzed in essen- tially the same way as that used here for Dirac electrons. 178 PHYSICAL REVIEW VOLUME 76, NUMBER 6 SEPTEMBER Space-Time Approach to Quantum Electrodynamics R. P. Feynman Department of Physics, Cornell University, Ithaca, New York (Received May 9, 1949) In this paper two things are done. (1) It is shown that a con- siderable simplification can be attained in writing down matrix elements for complex processes in electrodynamics. Further, a physical point of view is available which permits them to be written down directly for any specific problem. Being simply a restatement of conventional electrodynamics, however, the matrix elements diverge for complex processes. (2) Electrodynamics is modified by altering the interaction of electrons at short distances. All matrix elements are now finite, with the exception of those relating to problems of vacuum polarization. The latter are evaluated in a manner suggested by Pauli and Bethe, which gives finite results for these matrices also. The only effects sensitive to the modification are changes in mass and charge of the electrons. Such changes could not be directly observed. Phenomena directly observable, are insensitive to the details of the modification used (except at extreme energies). For such phenomena, a limit can be taken as the range of the modification goes to zero. The results then agree with those of Schwinger. A complete, unambiguous, and presumably consistent, method is therefore available for the calculation of all processes involving electrons and photons. The simplification in writing the expressions results from an emphasis on the over-all space-time view resulting from a study of the solution of the equations of electrodynamics. The relation of this to the more conventional Hamiltonian point of view is discussed. It would be very difficult to make the modification which is proposed if one insisted on having the equations in Hamiltonian form. The methods apply as well to charges obeying the Klein-Gordon equation, and to the various meson theories of nuclear forces. Illustrative examples are given. Although a modification like that used in electrodynamics can make all matrices finite for all of the meson theories, for some of the theories it is no longer true that all directly observable phenomena are insensitive to the details of the modification used. The actual evaluation of integrals appearing in the matrix elements may be facilitated, in the simpler cases, by methods described in the appendix. I HIS paper should be considered as a direct con- ■*■ tinuation of a preceding one 1 (I) in which the motion of electrons, neglecting interaction, was ana- lyzed, by dealing directly with the solution of the Hamiltonian differential equations. Here the same tech- nique is applied to include interactions and in that way to express in simple terms the solution of problems in quantum electrodynamics. For most practical calculations in quantum electro- dynamics the solution is ordinarily expressed in terms of a matrix element. The matrix is worked out as an expansion in powers of e 2 /hc, the successive terms cor- responding to the inclusion of an increasing number of virtual quanta. It appears that a considerable simplifi- cation can be achieved in writing down these matrix elements for complex processes. Furthermore, each term in the expansion can be written down and understood directly from a physical point of view, similar to the space-time view in I. It is the purpose of this paper to describe how this may be done. We shall also discuss methods of handling the divergent integrals which appear in these matrix elements. The simplification in the formulae results mainly from the fact that previous methods unnecessarily separated into individual terms processes that were closely related physically. For example, in the exchange of a quantum between two electrons there were two terms depending on which electron emitted and which absorbed the quantum. Yet, in the virtual states considered, timing relations are not significant. Olny the order of operators in the matrix must be maintained. We have seen (I), that in addition, processes in which virtual pairs are produced can be combined with others in which only 1 R. P. Feynman, Phys. Rev. 76, 749 (1949), hereafter called I. positive energy electrons are involved. Further, the effects of longitudinal and transverse waves can be combined together. The separations previously made were on an unrelativistic basis (reflected in the circum- stance that apparently momentum but not energy is conserved in intermediate states). When the terms are combined and simplified, the relativistic invariance of the result is self-evident. We begin by discussing the solution in space and time of the Schrodinger equation for particles interacting instantaneously. The results are immediately general- izable to delayed interactions of relativistic electrons and we represent in that way the laws of quantum electrodynamics. We can then see how the matrix ele- ment for any process can be written down directly. In particular, the self-energy expression is written down. So far, nothing has been done other than a restate- ment of conventional electrodynamics in other terms. Therefore, the self-energy diverges. A modification 2 in interaction between charges is next made, and it is shown that the self-energy is made convergent and corresponds to a correction to the electron mass. After the mass correction is made, other real processes are finite and insensitive to the "width" of the cut-off in the interaction. 3 Unfortunately, the modification proposed is not com- pletely satisfactory theoretically (it leads to some diffi- culties of conservation of energy). It does, however, seem consistent and satisfactory to define the matrix 2 For a discussion of this modification in classical physics see R. P. Feynman, Phys. Rev. 74 939 (1948), hereafter referred to as A. 3 A brief summary of the methods and results will be found in R. P. Feynman, Phys. Rev. 74, 1430 (1948), hereafter referred to as B. 769 179 770 FEYNMAN element for all real processes as the limit of that com- puted here as the cut-off width goes to zero. A similar technique suggested by Pauli and by Bethe can be applied to problems of vacuum polarization (resulting in a renormalization of charge) but again a strict physical basis for the rules of convergence is not known. After mass and charge renormalization, the limit of zero cut-off width can be taken for all real processes. The results are then equivalent to those of Schwinger 4 who does not make explicit use of the convergence fac- tors. The method of Schwinger is to identify the terms corresponding to corrections in mass and charge and, previous to their evaluation, to remove them from the expressions for real processes. This has the advantage of showing that the results can be strictly independent of particular cut-off methods. On the other hand, many of the properties of the integrals are analyzed using formal properties of invariant propagation functions. But one of the properties is that the integrals are infinite and it is not clear to what extent this invalidates the demonstrations. A practical advantage of the present method is that ambiguities can be more easily resolved; simply by direct calculation of the otherwise divergent integrals. Nevertheless, it is not at all clear that the convergence factors do not upset the physical con- sistency of the theory. Although in the limit the two methods agree, neither method appears to be thoroughly satisfactory theoretically. Nevertheless, it does appear that we now have available a complete and definite method for the calculation of physical processes to any order in quantum electrodynamics. Since we can write down the solution to any physical problem, we have a complete theory which could stand by itself. It will be theoretically incomplete, however, in two respects. First, although each term of increasing order in e 2 /hc can be written down it would be desirable to see some way of expressing things in finite form to all orders in e 2 /hc at once. Second, although it will be physically evident that the results obtained are equiva- lent to those obtained by conventional electrodynamics the mathematical proof of this is not included. Both of these limitations will be removed in a subsequent paper (see also Dyson 4 ). Briefly the genesis of this theory was this. The con- ventional electrodynamics was expressed in the La- grangian form of quantum mechanics described in the Reviews of Modern Physics. 5 The motion of the field oscillators could be integrated out (as described in Sec- tion 13 of that paper), the result being an expression of the delayed interaction of the particles. Next the modi- fication of the delta-function interaction could be made directly from the analogy to the classical case. 2 This 4 J. Schwinger, Phys. Rev. 74, 1439 (1948), Phys. Rev. 75, 651 (1949). A proof of this- equivalence is given by F. J. Dyson, Phys. Rev. 75, 486 (1949). 5 R. P. Feynman, Rev. Mod. Phys. 20, 367 (1948). The applica- tion to electrodynamics is described in detail by H. J. Groenewold, Koninklijke Nederlandsche Akademia van Weteschappen. Pro- ceedings Vol. LII, 3 (226) 1949. was still not complete because the Lagrangian method had been worked out in detail only for particles obeying the non-relativistic Schrodinger equation. It was then modified in accordance with the requirements of the Dirac equation and the phenomenon of pair creation. This was made easier by the reinterpretation of the theory of holes (I). Finally for practical calculations the expressions were developed in a power series in e 2 /hc. It was apparent that each term in the series had a simple physical interpretation. Since the result was easier to understand than the derivation, it was thought best to publish the results first in this paper. Considerable time' has been spent to make these first two papers as com- plete and as physically plausible as possible without relying on the Lagrangian method, because it is not generally familiar. It is realized that such a description cannot carry the conviction of truth which would ac- company the derivation. On the other hand, in the interest of keeping simple things simple the derivation will appear in a separate paper. The possible application of these methods to the various meson theories is discussed briefly. The formu- las corresponding to a charge particle of zero spin moving in accordance with the Klein Gordon equation are also given. In an Appendix a method is given for calculating the integrals appearing in the matrix ele- ments for the simpler processes. The point of view which is taken here of the inter- action of charges differs from the more usual point of view of field theory. Furthermore, the familiar Hamil- tonian form of quantum mechanics must be compared to the over-all space-time view used here. The first section is, therefore, devoted to a discussion of the relations of these viewpoints. 1. COMPARISON WITH THE HAMILTONIAN METHOD Electrodynamics can be looked upon in two equiva- lent and complementary ways. One is as the description of the behavior of a field (Maxwell's equations). The other is as a description of a direct interaction at a distance (albeit delayed in time) between charges (the solutions of Lienard and Wiechert). From the latter point of view light is considered as an interaction of the charges in the source with those in the absorber. This is an impractical point of view because many kinds of sources produce the same kind of effects. The field point of view separates these aspects into two simpler prob- lems, production of light, and absorption of light. On the other hand, the field point of view is less practical when dealing with close collisions of particles (or their action on themselves). For here the source and absorber are not readily distinguishable, there is an intimate exchange of quanta. The fields are so closely determined by the motions of the particles that it is just as well not to separate the question into two problems but to con- sider the process as a direct interaction. Roughly, the field point of view is most practical for problems involv- 180 QUANTUM ELECTRODYNAMICS 771 ing real quanta, while the interaction view is best for the discussion of the virtual quanta involved. We shall emphasize the interaction viewpoint in this paper, first because it is less familiar and therefore requires more discussion, and second because the important aspect in the problems with which we shall deal is the effect of virtual quanta. The Hamiltonian method is not well adapted to represent the direct action at a distance between charges because that action is delayed. The Hamiltonian method represents the future as developing out of the present. If the values of a complete set of quantities are known now, their values can be computed at the next instant in time. If particles interact through a delayed inter- action, however, one cannot predict the future by simply knowing the present motion of the particles. One would also have to know what the motions of the particles were in the past in view of the interaction this may have on the future motions. This is done in the Hamiltonian electrodynamics, of course, by requiring that one specify besides the present motion of the particles, the values of a host of new variables (the coordinates of the field oscillators) to keep track of that aspect of the past motions of the particles which de- termines their future behavior. The use of the Hamil- tonian forces one to choose the field viewpoint rather than the interaction viewpoint. In many problems, for example, the close collisions of particles, we are not interested in the precise tem- poral sequence of events. It is not of interest to be able to say how the situation would look at each instant of time during a collision and how it progresses from in- stant to instant. Such ideas are only useful for events taking a long time and for which we can readily obtain information during the intervening period. For collisions it is much easier to treat the process as a whole. 6 The M0ller interaction matrix for the the collision of two elec- trons is not essentially more complicated than the non- relativistic Rutherford formula, yet the mathematical machinery used to obtain the former from quantum electrodynamics is vastly more complicated than Schrodinger's equation with the e 2 /rn interaction needed to obtain the latter. The difference is only that in the latter the action is instantaneous so that the Hamiltonian method requires no extra variables, while in the former relativistic case it is delayed and the Hamiltonian method is very cumbersome. We shall be discussing the solutions of equations rather than the time differential equations from which they come. We shall discover that the solutions, because of the over-all space-time view that they permit, are as easy to understand when interactions are delayed as when they are instantaneous. As a further point, relativistic invariance will be self- evident. The Hamiltonian form of the equations de- velops the future from the instantaneous present. But 6 This is the viewpoint of the theory of the 5 matrix of Heisen- berg. for different observers in relative motion the instan- taneous present is different, and corresponds to a different 3-dimensional cut of space-time. Thus the temporal analyses of different observers is different and their Hamiltonian equations are developing the process in different ways. These differences are irrelevant, how- ever, for the solution is the same in any space time frame. By forsaking the Hamiltonian method, the wedding of relativity and quantum mechanics can be accomplished most naturally. We illustrate these points in the next section by studying the solution of Schrodinger's equation for non- relativistic particles interacting by an instantaneous Coulomb potential (Eq. 2). When the solution is modi- fied to include the effects of delay in the interaction and the relativistic properties of the electrons we obtain an expression of the laws of quantum electrodynamics (Eq. 4). 2. THE INTERACTION BETWEEN CHARGES We study by the same methods as in I, the interaction of two particles using the same notation as I. We start by considering the non-relativistic case described by the Schrodinger equation (I, Eq. 1). The wave function at a given time is a function ip(x a , x b , t) of the coordinates x a and x b of each particle. Thus call A'(x a , x b , /; x a ', x b , /') the amplitude that particle a at x a ' at time /' will get to x a at / while particle b at x b ' at t' gets to x fc at /. If the particles are free and do not interact this is A^(x a , x 6 , /; x a ', x b , i') = K 0a (x a , t; x a ', t')K 0b (x b , (; x b ', /') where K 0a is the A' function for particle a considered as free. In this case we can obviously define a quantity like K, but for which the time / need not be the same for particles a and b (likewise for t'); e.g., K (3, 4; 1, 2) = K 0a (3, 1)£ 06 (4, 2) (1) can be thought of as the amplitude that particle a goes from Xi at t\ to x 3 at / 3 and that particle b goes from x 2 at t 2 to x 4 at t A . When the particles do interact, one can only define the quantity K(3, 4; 1, 2) precisely if the interaction vanishes between ti and t% and also between tz and U. In a real physical system such is not the case. There is such an enormous advantage, however, to the concept that we shall continue to use it, imagining that we can neglect the effect of interactions between t x and h and between / 3 and t^ For practical problems this means choosing such long time intervals / 3 — h and i 4 — h that the extra interactions near the end points have small relative effects. As an example, in a scattering problem it may well be that the particles are so well separated initially and finally that the interaction at these times is negligible. Again energy values can be defined by the average rate of change o'f phase over such long time intervals that errors initially and finally can be neg- lected. Inasmuch as any physical problem can be defined in terms of scattering processes we do not lose much in 181 772 R.P.FEYNMAN Fig. 1. The fundamental interaction Eq. (4). Exchange of one quantum between two electrons. a general theoretical sense by this approximation. If it is not made it is not easy to study interacting particles relativistically, for there is nothing significant in choos- ing t x =tz if Xi^x 3 , as absolute simultaneity of events at a distance cannot be defined invariantly. It is essen- tially to avoid this approximation that the complicated structure of the older quantum electrodynamics has been built up. We wish to describe electrodynamics as a delayed interaction between particles. If we can make the approximation of assuming a meaning to K (3, 4; 1, 2) the results of this interaction can be expressed very simply. To see how this may be done, imagine first that the interaction is simply that given by a Coulomb potential #/r where r is the distance between the particles. If this be turned on only for a very short time A/ at time t , the first order correction to if (3, 4; 1, 2) can be worked out exactly as was Eq. (9) of I by an obvious general- ization to two particles: K»(3, 4; 1 ,2) = -ie 2 f f ' K 0a (3, 5)A' 0b (4, 6)r 66 " 1 XA' 0a (5, l)A' 0h (6, 2)<Z 3 x 5 d 3 x 6 A* , where t b = l 6 = t . If now the potential were on at all times (so that strictly K is not defined unless ti = t 3 and h=h), the first-order effect is obtained by integrating on to, which we can write as an integral over both t 6 and h if we include a delta- function 5(/ 5 — to) to insure contribution only when t b = t 6 . Hence, the first-order effect of interaction is (calling t b — t 6 =t b6 ): K™(3, 4; 1, 2)=-ie 2 f f K Qa (3, 5)K 0b (4, 6)r 66 - ] X5(t b ,)K 0a (5, l)K ob (6, 2)dT b dr 6 , (2) where dr = d 3 xdt. We know, however, in classical electrodynamics, that the Coulomb potential does not act instantaneously, but is delayed by a time r 56 , taking the speed of light as unity. This suggests simply replacing r 6 6 _1 5(/56) in (2) by something like r 5 6 _1 5(/ 5 6— r b6 ) to represent the delay in the effect of b on a. This turns out to be not quite right, 7 for when this interaction is represented by photons they must be of only positive energy, while the Fourier transform of S(t b 6— r b6 ) contains frequencies of both signs. It should instead be replaced by 5+(/ 56 — r 56 ) where 5+0*0= I e-' ux du/TT = \im =5(x)-\-( J Q «~o x-ie (3) This is to be averaged with r 56 -1 5+(— / 5 6— r 5 e) which arises when t b <lo and corresponds to a emitting the quantum which b receives. Since (2r)->(5+0-r) + 5+(-/-r)) = 5 + (/ 2 -r 2 ), this means r bf r l 8(t b e) is replaced by 5 + (s 56 2 ) where s b o 2 = t bb 2 — r bf ? is the square of the relativistically in- variant interval between points 5 and 6. Since in classical electrodynamics there is also an interaction through the vector potential, the complete interaction (see A, Eq. (1)) should be (1 — (v 5 -v 6 )5+(5 5 6 2 ), or in the relativistic case, (1 — a a - u b )5 + (s b <f) = (3 a j3 b y aiJi y bli 5 + (s b <r). Hence we have for electrons obeying the Dirac equation, KW(3, 4; 1,2) = - ie 2 J> ,(3,5)A' +b (4,6) ToM7 ^ XS+(*56 2 )£+a(S, l)A' +b (6, 2)dr s dTo, (4) where y ai i and 7 hM are the Dirac matrices applying to the spinor corresponding to particles a and b, respec- tively (the factor (3 a (3 b being absorbed in the definition, I Eq. (17), of K+). This is our fundamental equation for electrodynamics. It describes the effect of exchange of one quantum (therefore first order in e 2 ) between two electrons. It will serve as a prototype enabling us to write down the corresponding quantities involving the exchange of two or more quanta between two electrons or the interaction of an electron with itself. It is a consequence of con- ventional electrodynamics. Relativistic invariance is clear. Since one sums over // it contains the effects of both longitudinal and transverse waves in a relati- vistically symmetrical way. We shall now interpret Eq. (4) in a manner which will permit us to write down the higher order terms. It can be understood (see Fig. 1) as saying that the ampli- tude for "a" to go from 1 to 3 and "6" to go from 2 to 4 is altered to first order because they can exchange a quantum. Thus, "a" can go to 5 (amplitude A^+(5, 1)) 7 It, and a like term for the effect of a on b, leads to a theory which, in the classical limit, exhibits interaction through half- advanced and half-retarded potentials. Classically, this is equi- valent to purely retarded effects within a closed box from which no light escapes (e.g., see A, or J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945)). Analogous theorems exist in quantum mechanics but it would lead us too far astray to discuss them now. 182 QUANTUM ELECTRODYNAMICS 773 emit a quantum (longitudinal, transverse, or scalar y ail ) and then proceed to 3 (K + (3, 5)). Meantime "b" goes to 6 (K+(6, 2)), absorbs the quantum (yb») and proceeds to 4 (#+(4, 6)). The quantum meanwhile pro- ceeds from 5 to 6, which it does with amplitude 5 + (s 66 2 ). We must sum over all the possible quantum polariza- tions n and positions and times of emission 5, and of absorption 6. Actually if t b >t 6 it would be better to say that "a" absorbs and "b" emits but no attention need be paid to these matters, as all such alternatives are automatically contained in (4). The correct terms of higher order in e 2 or involving larger numbers of electrons (interacting with themselves or in pairs) can be written down by the same kind of reasoning. They will be illustrated by examples as we proceed. In a succeeding paper they will all be deduced from conventional quantum electrodynamics. Calculation, from (4), of the transition element be- tween positive energy free electron states gives the Moller scattering of two electrons, when account is taken of the Pauli principle. The exclusion principle for interacting charges is handled in exactly the same way as for non-interacting charges (I). For example, for two charges it requires only that one calculate K(3, 4; 1, 2) -#(4, 3; 1,2) to get the net amplitude for arrival of charges at 3 and 4. It is disregarded in intermediate states. The inter- ference effects for scattering of electrons by positrons discussed by Bhabha will be seen to result directly in this formulation. The formulas are interpreted to apply to positrons in the manner discussed in I. As our primary concern will be for processes in which the quanta are virtual we shall not include here the detailed analysis of processes involving real quanta in initial or final state, and shall content ourselves by only stating the rules applying to them. 8 The result of the analysis is, as expected, that they can be included by the same line of reasoning as is used in discussing the virtual processes, provided the quantities are normalized in the usual manner to represent single quanta. For example, the amplitude that an electron in going from 1 to 2 absorbs a quantum whose vector potential, suitably normalized, is c M exp(— ik-x) = C li (x) is just the expres- sion (I, Eq. (13)) for scattering in a potential with A (3) replaced by C (3). Each quantum interacts only 8 Although in the expressions stemming from (4) the quanta are virtual, this is not actually a theoretical limitation. One way to deduce the correct rules for real quanta from (4) is to note that in a closed system all quanta can be considered as virtual (i.e., they have a known source and are eventually absorbed) so that in such a system the present description is complete and equiva- lent to the conventional one. In particular, the relation of the Einstein A and B coefficients can be deduced. A more practical direct deduction of the expressions for real quanta will be given in the subsequent paper. It might be noted that (4) can be re- written as describing the action on a, K a) (3, l) = iJ'K + (3, 5) XA(5)K+(5, \)dn of the potential 4 M (5) = e 2 /"A' + (4, 6)8 + (s M i )y )1 XK + (6,2)dTt arising from Maxwell's equations — O i A ll = iirj ll from a "current" Jn(6) =e?K + (4, 6)y^K+(6, 2) produced by par- ticle b in going from 2 to 4. This is virtue of the fact that 8 + satisfies - D^+^i 2 ) = 4*5(2, 1). (5) once (either in emission or in absorption), terms like (I, Eq. (14)) occur only when there is more than one quantum involved. The Bose statistics of the quanta can, in all cases, be disregarded in intermediate states. The only effect of the statistics is to change the weight of initial or final states. If there are among quanta, in the initial state, some n which are identical then the weight of the state is (1/w!) of what it would be if these quanta were considered as different (similarly for the final state). 3. THE SELF-ENERGY PROBLEM Having a term representing the mutual interaction of a pair of charges, we must include similar terms to represent the interaction of a charge with itself. For under some circumstances what appears to be two dis- tinct electrons may, according to I, be viewed also as a single electron (namely in case one electron was created in a pair with a positron destined to annihilate the other electron). Thus to the interaction between such electrons must correspond the possibility of the action of an electron on itself. 9 This interaction is the heart of the self energy prob- lem. Consider to first order in e 2 the action of an electron on itself in an otherwise force free region. The amplitude K(2, 1) for a single particle to get from 1 to 2 differs from K + {2, 1) to first order in e 2 by a term k«(2, 1)= -ie 2 f lK + (2, 4) 7m # + (4, 3J 7 „ XK+(3, l)dT- 3 d u 5,(s iZ 2 ). (6) It arises because the electron instead of going from 1 directly to 2, may go (Fig. 2) first to 3, (K + (3, 1)), emit a quantum ( 7/( ), proceed to 4, (if+(4, 3)), absorb it ( 7m ), and finally arrive at 2 (K + (2, 4)). The quantum must go from 3 to 4 (5 + (s 4 3 2 )). This is related to the self-energy of a free electron in the following manner. Suppose initially, time t h we have an electron in state /(l) which we imagine to be a posi- tive energy solution of Dirac's equation for a free par- ticle. After a long time h—t\ the perturbation will alter Fig. 2. Interaction of an elec- tron with itself, Eq. (6). 9 These considerations make it appear unlikely that the con- tention of J. A. Wheeler and R. P. Feynman, Rev. Mod. Phys. 17, 157 (1945), that electrons do not act on themselves, will be a successful concept in quantum electrodynamics. 183 774 P. FEYNMAN the wave function, which can then be looked upon as a superposition of free particle solutions (actually it only contains /). The amplitude that g(2) is contained is calculated as in (I, Eq. (21)). The diagonal element (g=f) is therefore //• f(2)(3KU(2, 1)^/(1)^x^X2 (7) The time interval T = l 2 —t 1 (and the spatial volume V over which one integrates) must be taken very large, for the expressions are only approximate (analogous to the situation for two interacting charges). 10 This is because, for example, we are dealing incorrectly with quanta emitted just before t 2 which would normally be reabsorbed at times after t 2 . If K m (2, 1) from (6) is actually substituted into (7) the surface integrals can be performed as was done in obtaining I, Eq. (22) resulting in *//■ /(4) Tm K + (4, 3h li f(3)8 + (s i J)dudu. (8) Putting for /(l) the plane wave u exp(— ip-xi) where pt is the energy (p 4 ) and momentum of the electron (J) 2 = m 2 ), and u is a constant 4-index symbol, (8) becomes ■te 2 ff(u yii K + (4, 3) 7 „«) X exp(i/> • (x 4 — x 3 )) 8 + (s i3 2 )dT 3 dTi, the integrals extending over the volume V and time interval T. Since if + (4, 3) depends only on the difference of the coordinates of 4 and 3, x 43m , the integral on 4 gives a result (except near the surfaces of the region) independent of 3. When integrated on 3, therefore, the result is of order VT. The effect is proportional to V, for the wave functions have been normalized to unit MOMENTUM p-k, FACTOR (p-k-m) INTERACTION ,lL MOMENTUM k, FACTOR k~ 2 MOMENTUM p Fig. 3. Interaction of an electron with itself. Momentum space, Eq. (11). 10 This is discussed in reference 5 in which it is pointed out that the concept of a wave function loses accuracy if there are delayed self-actions. volume. If normalized to volume V, the result would simply be proportional to T. This is expected, for if the effect were equivalent to a change in energy AE, the amplitude for arrival in / at t 2 is altered by a factor exp( — iAE{t 2 — ti)), or to first order by the difference — i(AE)T. Hence, we have AE = e 2 I («7 M iiC + (4, 3)7^) exp(^-x«)5 + (543 2 )rfT 4 , (9) integrated over all space-time dr 4 . This expression will be simplified presently. In interpreting (9) we have tacitly assumed that the wave functions are normalized so that (u*u)= (uyiu) = l. The equation may therefore be made independent of the normalization by writing the left side as (AE)(uyw), or since (uyiu) = (E/m)(uu) and mAm = EAE, as Am(uu) where Am is an equivalent change in mass of the electron. In this form invariance is obvious. One can likewise obtain an expression for the energy- shift for an electron in a hydrogen atom. Simply replace K+ in (8), by K+ iV) , the exact kernel for an electron in the potential, V=fie 2 /r, of the atom, and / by a wave function (of space and time) for an atomic state. In general the AE which results is not real. The imaginary part is negative and in exp( — iAET) produces an ex- ponentially decreasing amplitude with time. This is because we are asking for the amplitude that an atom initially with no photon in the field, will still appear after time T with no photon. If the atom is in a state which can radiate, this amplitude must decay with time. The imaginary part of AE when calculated does indeed give the correct rate of radiation from atomic states. It is zero for the ground state and for a free electron. In the non-relativistic region the expression for AE can be worked out as has been done by Bethe. 11 In the relativistic region (points 4 and 3 as close together as a Compton wave-length) the K + (V) which should appear in (8) can be replaced to first order in V by K + plus K+ (l) (2, 1) given in I, Eq. (13). The problem is then very similar to the radiationless scattering problem discussed below. 4. EXPRESSION IN MOMENTUM AND ENERGY SPACE The evaluation of (9), as well as all the other more complicated expressions arising in these problems, is very much simplified by working in the momentum and energy variables, rather than space and time. For this we shall need the Fourier Transform of 5 + (s 2 i 2 ) which is 5 + (W): 1 I exp(— ik- x 21 )k- 2 d% (10) which can be obtained from (3) and (5) or from I, Eq. (32) noting that 7+(2, 1) for m 2 = is 5+(s 21 2 ) from 11 H. A. Bethe, Phys. Rev. 72, 339 (1947). 184 444W4-W4WWW44444W44444-4W4WW4-44-4-W4-444- 4"9"9-4- 4-9-&J-4. 4.4. QUANTUM ELECTRODYNAMICS 775 a. Eql2 b. Eql3 c. Eq.14 Fig. 4. Radiative correction to scattering, momentum space. I, Eq. (34). The k~ 2 means (k-k)' 1 or more precisely the limit as 5— >0 of (k-k+id)' 1 . Further d 4 k means (liry-dkidk-idkzdki. If we imagine that quanta are par- ticles of zero mass, then we can make the general rule that all poles are to be resolved by considering the masses of the particles and quanta to have infinitesimal negative imaginary parts. Using these results we see that the self-energy (9) is the matrix element between u and u of the matrix {e 2 /iri) jy,(p- ■k-m)- i y l JrH% en: where we have used the expression (I, Eq. (31)) for the Fourier transform of K + . This form for the self-energy is easier to work with than is (9). The equation can be understood by imagining (Fig. 3) that the electron of momentum p emits (7^) a quantum of momentum k, and makes its way now with mo- mentum p— k to the next event (factor {p—k—m)" 1 ) which is to absorb the quantum (another 7 M ). The amplitude of propagation of quanta is k~ 2 . (There is a factor e 2 /iri for each virtual quantum). One integrates over all quanta. The reason an electron of momentum p propagates as \/(p—m) is that this operator is the re- ciprocal of the Dirac equation operator, and we are simply solving this equation. Likewise light goes as l/k 2 , for this is the reciprocal D'Alembertian operator of the wave equation of light. The first y„ represents the current which generates the vector potential, while the second is the velocity operator by which this poten- tial is multiplied in the Dirac equation when an external field acts on an electron. Using the same line of reasoning, other problems may be set up directly in momentum space. For example, consider the scattering in a potential A = A ll y^ varying in space and time as a exp(-iq-x). An electron initially in state of momentum pi = pi^.y li will be deflected to state pi where Pi = pi-\-q. The zero-order answer is simply the matrix element of a between states 1 and 2. We next ask for the first order (in e 2 ) radiative correc- tion due to virtual radiation of one quantum. There are several ways this can happen. First for the case illus- (a) (b) Fig. 5. Compton scattering, Eq. (15). trated in Fig. 4(a), find the matrix: {e 2 /m) fy^po-k-my'a^-k-my^y^k-Wk. (12) For in this case, first 12 a quantum of momentum k is emitted (7 M ), the electron then having momentum pi— k and hence propagating with factor {pi—k—m)~ l . Next it is scattered by the potential (matrix a) receiving additional momentum q, propagating on then (factor (p2—k—m)~ l ) with the new momentum until the quan- tum is reabsorbed (7^). The quantum propagates from emission to absorption (k~ 2 ) and we integrate over all quanta (d 4 k), and sum on polarization fi. When this is integrated on & 4 , the result can be shown to be exactly equal to the expressions (16) and (17) given in B for the same process, the various terms coming from resi- dues of the poles of the integrand (12). Or again if the quantum is both emitted and re- absorbed before the scattering takes place one finds (Fig. 4(b)) »/ (e 2 /iri) a(/> 1 -.ra)- 1 7„(£i-fe-w)- 1 7„fe- 2 (2 4 Jk, (13) or if both emission - and absorption occur after the scattering, (Fig. 4(c)) (e 2 A0 f 7„(/> 2 - k-my^ipi- ~ l ak~ 2 d 4 k. (14) These terms are discussed in detail below. We have now achieved our simplification of the form of writing matrix elements arising from virtual proc- esses. Processes in which a number of real quanta is given initially and finally offer no problem (assuming correct normalization). For example, consider the Compton effect (Fig. 5(a)) in which an electron in state pi absorbs a quantum of momentum q h polarization vector ei M so that its interaction is £ , i M 7 M = ei, and emits a second quantum of momentum —q», polarization e 2 to arrive in final state of momentum pi. The matrix for 12 First, next, etc., here refer not to the order in true time but to the succession of events along the trajectory of the electron. That is, more precisely, to the order of appearance of the matrices in the expressions. 185 776 FEYNMAN this process is e 2 (pi+qi— m)~ l e\. The total matrix for the Compton effect is, then, e2(pi+qi-my i e 1 +e 1 (J) 1 +q2-m)- 1 e2, (15) the second term arising because the emission of e 2 may also precede the absorption of e x (Fig. 5(b)). One takes matrix elements of this between initial and final electron states (pi+qi = p2— 92), to obtain the Klein Nishina formula. Pair annihilation with emission of two quanta, etc., are given by the same matrix, positron states being those with negative time component of p. Whether quanta are absorbed or emitted depends on whether the time component of q is positive or negative. 5. THE CONVERGENCE OF PROCESSES WITH VIRTUAL QUANTA These expressions are, as has been indicated, no more than a re-expression of conventional quantum electro- dynamics. As a consequence, many of them are mean- ingless. For example, the self-energy expression (9) or (11) gives an infinite result when evaluated. The infinity arises, apparently, from the coincidence of the 5-function singularities in K + (A, 3) and 8 + (sw 2 )- Only at this point is it necessary to make a real departure from conven- tional electrodynamics, a departure other than simply rewriting expressions in a simpler form. We desire to make a modification of quantum electro- dynamics analogous to the modification of classical electrodynamics described in a previous article, A. There the 8(si 2 2 ) appearing in the action of interaction was replaced by f(si 2 2 ) where f(x) is a function of small width and great height. The obvious corresponding modification in the quan- tum theory is to replace the 8 + (s 2 ) appearing the quantum mechanical interaction by a new function f+(s 2 ). We can postulate that if the Fourier trans- form of the classical /(S12 2 ) is the integral over all k of F{k 2 ) exp(—ik -Xi2)d 4 k, then the Fourier transform of f + (s 2 ) is the same integral taken over only positive fre- quencies ki for t 2 >ti and over only negative ones for h<h in analogy to the relation of 8+(s 2 ) to 8(s 2 ). The function f(s 2 ) = f(x-x) can be written* as f(x-x) = (2ir)- 2 C f sin(£ 4 |* 4 |) Xcos(K-x)dk i d 3 Kg(k- k), where g{k ■ k) is kc 1 times the density of oscillators and may be expressed for positive h as (A, Eq. (16)) i(k 2 )= f (8(k 2 )-8(k 2 -\ 2 ))G(\)d\ where Jl x G(X)d\=l and G involves values of X large compared to m. This simply means that the amplitude * This relation is given incorrectly in A, equation just pre- ceding 16. for propagation of quanta of momentum k is - F + (k 2 ) = 7T- 1 r (ft- 2 - (k 2 - \ 2 )- i )G(\)d\, Jo rather than ft" 2 . That is, writing F+{k 2 ) = -ir- l kr 2 C(k 2 ), -f+(s 12 2 ) = T- 1 f exp(-^-x 12 )/j- 2 C'(ft 2 )rf 4 £. (16) Every integral over an intermediate quantum which previously involved a factor d 4 k/k 2 is now supplied with a convergence factor C{k 2 ) where C(ft 2 ) = t: ■\ 2 (k 2 -X 2 )- l G(X)d\. (17) The poles are defined by replacing k 2 by k 2 +i8 in the limit 5— >0. That is X 2 may be assumed to have an infini- tesimal negative imaginary part. The function f+(sn 2 ) may still have a discontinuity in value on the light cone. This is of no influence for the Dirac electron. For a particle satisfying the Klein Gordon equation, however, the interaction involves gradients of the potential which reinstates the 8 func- tion if / has discontinuities. The condition that / is to have no discontinuity in value on the light cone implies k 2 C(k 2 ) approaches zero as k 2 approaches infinity. In terms of G(X) the condition is J X 2 G(\)d\ = 0. (18) This condition will also be used in discussing the con- vergence of vacuum polarization integrals. The expression for the self-energy matrix is now {#!■ ■i) fy»(p-k-m)- l y»k- 2 d*kC(k 2 ), (19) which, since C(k 2 ) falls off at least as rapidly as 1/k 2 , converges. For practical purposes we shall suppose hereafter that C(k 2 ) is simply — \ 2 /(k 2 — X 2 ) implying that some average (with weight G(\)d\) over values of X may be taken afterwards. Since in all processes the quantum momentum will be contained in at least one extra factor of the form (p—k—m) -1 representing propagation of an electron while that quantum is in the field, we can expect all such integrals with their convergence factors to converge and that the result of all such processes will now be finite and definite (ex- cepting the processes with closed loops, discussed below, in which the diverging integrals are over the momenta of the electrons rather than the quanta). The integral of ( 1 9) with C ( ft 2 ) = - X 2 ( k 2 - X 2 ) - 1 noting that p 2 — m 2 , X^>m and dropping terms of order m/X, is (see Appendix A) (e 2 /27r)[4m(ln(X/m)+|)-i!>(ln(X/m) + 5/4)]. (20) 186 QUANTUM ELECTRODYNAMICS 777 When applied to a state of an electron of momentum p satisfying pii = mu, it gives for the change in mass (as in B, Eq. (9)) Aw = m0 2 /27r)(31n(\/w)+!). (21) 6. RADIATIVE CORRECTIONS TO SCATTERING We can now complete the discussion of the radiative corrections to scattering. In the integrals we include the convergence factor C(k 2 ), so that they converge for large k. Integral (12 I is also not convergent because of the well-known infra-red catastrophy. For this reason we calculate (as discussed in B) the value of the integral assuming the photons to have a small mass X m i n <^;ra<^X. The integral (12) becomes («y« i)j:7*(P*- k—m)~ l a{pi— k—m)' Xy,(k 2 - X min 2 )- 1 ^C(fe 2 - X min 2 ), which when integrated (see Appendix B) gives (e 2 /2r) times I 2( In— 1 V 1 ) +6 tanfl L V X min /\ tan20/ 4 r° -I H I a tanaJa a tan20 J J 1 29 + — (qa-aq) +ra, (22) 4m sin20 where {q 2 )* = 2m sin# and we have assumed the matrix to operate between states of momentum pi and pi = p\-\-q and have neglected terms of order X m i n /m, ra/X, and <7 2 /X 2 . Here the only dependence on the convergence factor is in the term ra, where r = ln(A/w)+9/4-21n(m/X min ). (23) As we shall see in a moment, the other terms (13), (14) give contributions which just cancel the ra term. The remaining terms give for small q, / 1 4g- / m 3\\ ■ 2 /47r)( —(qa-aq)+—a( In ) ) \2m 3m 2 \ X min 8// (24) which shows the change in magnetic moment and the Lamb shift as interpreted in more detail in B. 13 13 That the result given in B in Eq. (19) was in error was re- peatedly pointed out to the author, in private communication, by V. F. Weisskopf and J. B. French, as their calculation, com- pleted simultaneously with the author's early in 1948, gave a different result. French has finally shown that although the ex- pression for the radiationless scattering B, Eq. (18) or (24) above is correct, it was incorrectly joined onto Bethe's non-relativistic result. He shows that the relation ln2& ma x— l = lnX m i n used by the author should have been ln2£ ma x— 5/6=lnX m i n . This results in adding a term —(1/6) to the logarithm in B, Eq. (19) so that the result now agrees with that of J. B. French and V. F. Weisskopf, We must now study the remaining terms (13) and (14). The integral on k in (13) can be performed (after multiplication by C{k 2 )) since it involves nothing but the integral (19) for the self-energy and the result is- allowed to operate on the initial state tt\, (so that piU\ = muy). Hence the factor following a{pi — m)~ l wilt be just Am. But, if one now tries to expand \/{p\— w) = (pi+m)/(pi 2 —m 2 ) one obtains an infinite result, since pi 2 = m 2 . This is, however, just what is expected physically. For the quantum can be emitted and ab- sorbed at any time previous to the scattering. Such a process has the effect of a change in mass of the electron in the state 1. It therefore changes the energy by AE and the amplitude to first order in AE by —iAE-t where t is the time it is acting, which is infinite. That is, the major effect of this term would be canceled by the effect of change of mass Am. The situation can be analyzed in the following manner. We suppose that the electron approaching the scattering potential a has not been free for an infinite time, but at some time far past suffered a scattering by a potential b. If we limit our discussion to the effects of Ara and of the virtual radiation of one quantum be- tween two such scatterings each of the effects will be finite, though large, and their difference is determinate. The propagation from b to a is represented by a matrix a{j)'—m)~ l b, (25) in which one is to integrate possibly over p' (depending on details of the situation). (If the time is long between b and a, the energy is very nearly determined so that p' 2 is very nearly m 2 .) We shall compare the effect on the matrix (25) of the virtual quanta and of the change of mass Am. The effect of a virtual quantum is {er/m) I a(p' — w) _1 7 M (/>'— k — m)~ x Xjnip'-my'bk-WkCik 2 ), (26) while that of a change of mass can be written a(p'-m)- l Am(p'-m)- l b, (27) and we are interested in the difference (26)-(27). A simple and direct method of making this comparison is just to evaluate the integral on k in (26) and subtract from the result the expression (27) where Am is given in (21). The remainder can be expressed as a multiple — r(p' 2 ) of the unperturbed amplitude (25); -r(p'*)a(p'-m)- l b. (28) This has the same result (to this order) as replacing the potentials a and b in (25) by (1 — \r(p" 2 ))a and Phys. Rev. 75, 1240 (1949) and N. H. Kroll and W. E. Lamb, Phys. Rev. 75, 388 (1949). The author feels unhappily responsible for the very considerable delay in the publication of French's result occasioned by this error. This footnote is appropriately numbered. 187 778 P. FE YN M AN {\-\r{p' 2 ))b. In the limit, then, as p' 2 -+m 2 the net effect on the scattering is — \ra where r, the limit of r(J>' 2 ) as p' 2 —>m 2 (assuming the integrals have an infra- red cut-off), turns out to be just equal to that given in (23). An equal term — \ra arises from virtual transitions after the scattering (14) so that the entire ra term in (22) is canceled. The reason that r is just the value of (12) when q 2 = can also be seen without a direct calculation as follows: Let us call p the vector of length m in the direction of p' so that if />' 2 = w(l+e) 2 we have p'= (l+e)p and we take e as very small, being of order T~ l where T is the time between the scatterings b and a. Since (/>'— ra) -1 = (p'+m)/(p' 2 -m 2 )~(p+m)/2m 2 e, the quantity (25) is of order € _1 or T. We shall compute corrections to it only to its own order (e -1 ) in the limit e— >0. The term (27) can be written approximately 14 as (e 2 /iri) I a(p'— w) _1 7 M (/>— k— w) _1 Xy,{P'-m)- l bk- 2 d'kC{k 2 ), using the expression (19) for Am. The net of the two effects is therefore approximately 15 - (e 2 /iri) laip'-m)-^^- k-m)~ l ep(p- k-m)~ l X y.ip'-m^bk-WkCik 2 ), a term now of order 1/e (since {p'—m)~ l ^{p-\-m) \(2ra 2 e) _1 ) and therefore the one desired in the limit. Comparison to (28) gives for r the expression (J>i+m/2m) I y^pi-k-m^ipinr^ipi—k—m)-* X7„fe- 2 d 4 £C(& 2 ). (29) The integral can be immediately evaluated, since it is the same as the integral (12), but with ^ = 0, for a replaced by Pi/m. The result is therefore r-(pi/m) which when acting on the state U\ is just r, as p\U\ = mu\. For the same reason the term {pi-\-m)/2m in (29) is effectively 1 and we are left with — r of (23). 16 In more complex problems starting with a free elec- 14 The expression is not exact because the substitution of Aw by the integral in (19) is valid only if p operates on a state such that p can be replaced by m. The error, however, is of order a{p'-m)~ 1 (p—m)(J)'-m)~ 1 b which is a((l+e)p+m) (p-m) X((l+t)p+m)p(2e+e i )- i m- i . But since ^=«',we ha.vep(p-m) = —m{p—m) = (p—m)p so the net result is approximately a(p—m)b/Am i and is not of order 1/e but smaller, so that its effect drops out in the limit. 15 We have used, to first order, the general expansion (valid for any operators A, B) (A+B^^A-i-A^BA-i+A-tBA-iBA-i with A=p— k— m and B=p'—p = tp to expand the difference of (/>'- k-m)- 1 and (p- k-m)~ l . 16 The renormalization terms appearing B, Eqs. (14), (15) when translated directly into the present notation do not give twice (29) but give this expression with the central pim~ l factor replaced by my t /Ei where E\ = p\ v . for /x = 4. When integrated it therefore gives ra((pi+m)/2m)(my t /Ei) or ra~ra(myi/Ei)(pi-m)/2m. (Since piyt+y t Pi = 2Ei) which gives just ra, since p\Ui = mui. tron the same type of term arises from the effects of a virtual emission and absorption both previous to the other processes. They, therefore, simply lead to the same factor r so that the expression (23) may be used directly and these renormalization integrals need not be computed afresh for each problem. In this problem of the radiative corrections to scatter- ing the net result is insensitive to the cut-off. This means, of course, that by a simple rearrangement of terms previous to the integration we could have avoided the use of the convergence factors completely (see for example Lewis 17 ). The problem was solved in the manner here in order to illustrate how the use of such convergence factors, even when they are actually un- necessary, may facilitate analysis somewhat by remov- ing the effort and ambiguities that may be involved in trying to rearrange the otherwise divergent terms. The replacement of 5+ by /+ given in (16), (17) is not determined by the analogy with the classical prob- lem. In the classical limit only the real part of 8+ (i.e., just 5) is easy to interpret. But by what should the imaginary part, l/(7ris 2 ), of 5 + be replaced? The choice we have made here (in defining, as we have, the location of the poles of (17)) is arbitrary and almost certainly incorrect. If the radiation resistance is calculated for an atom, as the imaginary part of (8), the result de- pends slightly on the function /+. On the other hand the light radiated at very large distances from a source is independent of /+. The total energy absorbed by distant absorbers will not check with the energy loss of the source. We are in a situation analogous to that in the classical theory if the entire / function is made to contain only retarded contributions (see A, Appendix). One desires instead the analogue of (F) Tet of A. This problem is being studied. One can say therefore, that this attempt to find a consistent modification of quantum electrodynamics is incomplete (see also the question of closed loops, below). For it could turn out that any correct form of /+ which will guarantee energy conservation may at the same time not be able to make the self-energy integral finite. The desire to make the methods of simplifying the calculation of quantum electrodynamic processes more widely available has prompted this publication before an analysis of the correct form for /+ is complete. One might try to take the position that, since the energy discrepancies discussed vanish in the limit X— >°o, the correct physics might be considered to be that obtained by letting X— >co after mass renormalization. I have no proof of the mathematical consistency of this procedure, but the presumption is very strong that it is satisfac- tory. (It is also strong that a satisfactory form for / + can be found.) 7. THE PROBLEM OF VACUUM POLARIZATION In the analysis of the radiative corrections to scatter- ing one type of term was not considered. The potential 17 H. W. Lewis, Phys. Rev. 73, 173 (1948). 188 QUANTUM ELECTRODYNAMICS 779 which we can assume to vary as a M exp(— iq-x) creates a pair of electrons (see Fig. 6), momenta p a , —pb. This pair then reannihilates, emitting a quantum q = pb—p a , which quantum scatters the original electron from state 1 to state 2. The matrix element for this process (and the others which can be obtained by rearranging the order in time of the various events) is - («*/«) (fl*y„«i) fsp[(p a +q-m)-i Xy.(pa~ m^yJd'Paqr'Ciq 2 )^. (30) This is because the potential produces the pair with amplitude proportional to a v y v , the electrons of mo- menta p a and — (p a -\- q) proceed from there to annihi- late, producing a quantum (factor y a ) which propagates (factor q~ 2 C(q 2 )) over to the other electron, by which it is absorbed (matrix element of y^ between states 1 and 2 of the original electron (w 2 7 M «i)). All momenta p a and spin states of the virtual electron are admitted, which means the spur and the integral on d*p a are calculated. One can imagine that the closed loop path of the positron-electron produces a current ^Jn = Jp (31) which is the source of the quanta which act on the second electron. The quantity J,„=-(<?/iri) f Sp[(P+q-myi Xy,{p-m)- l y^y,% (32) is then characteristic for this problem of polarization of the vacuum. One sees at once that J ^ diverges badly. The modifi- cation of 5 to / alters the amplitude with which the current jn will affect the scattered electron, but it can do nothing to prevent the divergence of the integral (32) and of its effects. One way to avoid such difficulties is apparent. From one point of view we are considering all routes by which a given electron can get from one region of space-time to another, i.e., from the source of electrons to the apparatus which measures them. From this point of view the closed loop path leading to (32) is unnatural. It might be assumed that the only paths of meaning are those which start from the source and work their way in a continuous path (possibly containing many time reversals) to the detector. Closed loops would be ex- cluded. We have already found that this may be done for electrons moving in a fixed potential. Such a suggestion must meet several questions, how- ever. The closed loops are a consequence of the usual hole theory in electrodynamics. Among other things, they are required to keep probability conserved. The probability that no pair is produced by a potential is Fig. 6. Vacuum polarization ef- fect on scattering, Eq. (30). p + a not unity and its deviation from unity arises from the imaginary part of 7 M „. Again, with closed loops ex- cluded, a pair of electrons once created cannot annihi- late one another again, the scattering of light by light would be zero, etc. Although we are not experimentally sure of these phenomena, this does seem to indicate that the closed loops are necessary. To be sure, it is always possible that these matters of probability con- servation, etc., will work themselves out as simply in the case of interacting particles as for those in a fixed potential. Lacking such a demonstration the presump- tion is that the difficulties of vacuum polarization are not so easily circumvented. 18 An alternative procedure discussed in B is to assume that the function K+(2, 1) used above is incorrect and is to be replaced by a modified function K + ' having no singularity on the light cone. The effect of this is to provide a convergence factor C(p 2 —m 2 ) for every inte- gral over electron momenta. 19 This will multiply the integrand of (32) by C{f- m 2 )C{{p+q) 2 -m 2 ), since the integral was originally 8(p a — pb+ q)d 4 p a d 4 p b and both p a and pb get convergence factors. The integral now converges but the result is unsatisfactory. 20 One expects the current (31) to be conserved, that is q»j»=0 or q li J liV =0. Also one expects no current if a„ is a gradient, or a v = q v times a constant. This leads to the condition / M „g„ = which is equivalent to g M / M „ = since J M » is symmetrical. But when the expression (32) is integrated with such convergence factors it does not satisfy this condition. By altering the kernel from K to another, K', which does not satisfy the Dirac equation we have lost the gauge invariance, its consequent cur- rent conservation and the general consistency of the theory. One can see this best by calculating J ^q, directly from (32). The expression within the spur becomes (P J rq—>n)~ 1 q(p—m)- 1 y ll which can be written as the difference of two terms: (/»— w) _1 7 M — (p+q— ?») -1 Y„- Each of these terms would give the same result if the integration d*p were without a convergence factor, for 18 It would be very interesting to calculate the Lamb shift accurately enough to be sure that the 20 megacycles expected from vacuum polarization are actually present. 19 This technique also makes self-energy and radiationless scat- tering integrals finite even without the modification of 5+ to/ + for the radiation (and the consequent convergence factor C(k : ) for the quanta). See B. 20 Added to the terms given below (33) there is a term l(\ 3 -2 / j.- + lq' ! )8 tu , for C{k 2 ) = -X^tf-X 2 )" 1 , which is not gauge invariant. (In addition the charge renormalization has — 7/6 added to the logarithm.) 189 780 R. P. FEYNMAN the first can be converted into the second by a shift of the origin of p, namely p' = p+q- This does not result in cancelation in (32) however, for the convergence factor is altered by the substitution. A method of making (32) convergent without spoiling the gauge invariance has been found by Bethe and by Pauli. The convergence factor for light can be looked upon as the result of superposition of the effects of quanta of various masses (some contributing nega- tively). Likewise if we take the factor C(p 2 —m 2 ) = -X i (p°—m 2 -X 2 )- i so that (p 2 -m 2 )- l C(p 2 -m 2 ) = (p 2 —m 2 )~ l —(p 2 —m 2 —X 2 )~ l we are taking the differ- ence of the result for electrons of mass m and mass (X 2 +w 2 )*. But we have taken this difference for each propagation between interactions with photons. They suggest instead that once created with a certain mass the electron should continue to propagate with this mass through all the potential interactions until it closes its loop. That is if the quantity (32), integrated over some finite range of p, is called J ^(m 2 ) and the corresponding quantity over the same range of p, but with m replaced by (ra 2 +X 2 )* is J iit {m 2 -\-X 1 ) we should calculate )-J li ,(m 2 +X 2 )^G(X)dX, (32') the function G(X) satisfying Jl x G(X)dX=l and Jl x G(X)X 2 dX = 0. Then in the expression for J M „ P the range of p integration can be extended to infinity as the integral now converges. The result of the integration using this method is the integral on dX over G(X) of (see Appendix C) X 2 Av p = e 2 / 1 X 2 — (qtf,-8^q 2 )[ In— t \ 3 m- [4m 2 +2q L 3q 2 V tan0/ 9JV (33) with q 2 = 4m 2 sin 2 0. The gauge invariance is clear, since q^q^q^—q^S^) = 0. Operating (as it always will) on a potential of zero divergence the (q»q v — b^q^a, is simply — q 2 a„ the D'Alembertian of the potential, that is, the current pro- ducing the potential. The term — f (ln(X 2 /ra 2 ))(^„ — q 2 8^) therefore gives a current proportional to the current producing the potential. This would have the same effect as a change in charge, so that we would have a difference A(e 2 ) between e 2 and the experimen- tally observed charge, e 2 -\-A(e 2 ), analogous to the dif- ference between m and the observed mass. This charge depends logarithmically on the cut-off, A(e 2 )/e 2 = — (2e 2 /3ir) h\(X/m). After this renormalization of charge is made, no effects will be sensitive to the cut-off. After this is done the final term remaining in (33), contains the usual effects 21 of polarization of the vacuum. 21 E. A. Uehling, Phys. Rev. 48, 55 (1935), R. Serber, Phys. Rev. 48, 49 (1935). It is zero for a free light quantum (q 2 ~0). For small q z it behaves as (2/\S)q l (adding — \ to the logarithm in the Lamb effect). For q 2 >(2m) 2 it is complex, the imaginary part representing the loss in amplitude re- quired by the fact that the probability that no quanta are produced by a potential able to produce pairs ((q 2 ) i >2m) decreases with time. (To make the neces- sary analytic continuation, imagine m to have a small negative imaginary part, so that (1 — q 2 /4 : m 2 ) i becomes — i(q 2 /4:m 2 — l) s as q 2 goes from below to above 4m 2 . Then d=ir/2-\-iu where sinhw = + (<7 2 /4w 2 — 1)*, and - l/tan0 = i tanhtt = + i (q 2 - Am 2 ) K<7 2 ) -5 -) Closed loops containing a number of quanta or poten- tial interactions larger than two produce no trouble. Any loop with an odd number of interactions gives zero (I, reference 9). Four or more potential interactions give integrals which are convergent even without a con- vergence factor as is well known. The situation is analogous to that for self-energy. Once the simple problem of a single closed loop is solved there are no further divergence difficulties for more complex processes. 22 8. LONGITUDINAL WAVES In the usual form of quantum electrodynamics the longitudinal and transverse waves are given separate treatment. Alternately the condition (dAJdx l ,) < if = Q is carried along as a supplementary condition. In the present form no such special considerations are neces- sary for we are dealing with the solutions of the equation — I \ 2 A u =4:irj u with a current _/ M which is conserved dj fl /dx^ = 0. That means at least □ ! (6M ;1 /dx M ) = and in fact our solution also satisfies 6M M /dx M =0. To show that this is the case we consider the ampli- tude for emission (real or virtual) of a photon and show that the divergence of this amplitude vanishes. The amplitude for emission for photons polarized in the ^ direction involves matrix elements of y m . Therefore what we have to show is that the corresponding matrix elements of q^y^q vanish. For example, for a first order effect we would require the matrix element of q between two states pi and p2=pr\-q. But since q = p2~pi and (u2piUi) = m(u2iii) = (u2p2Ui) the matrix element vanishes, which proves the contention in this case. It also vanishes in more complex situations (essen- tially because of relation (34), below) (for example, try putting e 2 = <72 in the matrix (15) for the Compton Effect). To prove this in general, suppose d, i= 1 to N are a set of plane wave disturbing potentials carrying mo- menta qi (e.g., some may be emissions or absorptions of the same or different quanta) and consider a matrix for the transition from a state of momentum p to px such 22 There are loops completely without external interactions. For example, a pair is created virtually along with a photon. Next they annihilate, absorbing this photon. Such loops are disregarded on the grounds that they do not interact .with anything and are thereby completely unobservable. Any indirect effects they may have via the exclusion principle have already been included. 190 QUANTUM ELECTRODYNAMICS 781 as as ITt=i Ar_1 (Pi— «*) _1 a,- where />,=/> i_i+ Qi (and in the product, terms with larger i are written to the left). The most general matrix element is simply a linear combination of these. Next consider the matrix be- tween states p and px+Q in a situation in which not only are the a* acting but also another potential a exp( — iq • x) where a = q. This may act previous to all a,, in which case it gives CLNWipi+q— m)~ l ai(p +q— m)~ l q which is equivalent to +aNYl(pi+q—m)~ l ai since -\-{po-\-q — m)~ l q is equivalent to (po+q— m)~ l Xipo+q—m) as p is equivalent to m acting on the initial state. Likewise if it acts after all the potentials it gives q{pN~ fn)~ l a N J\(pi— w) -l fl» which is equivalent to — ajvII(/ > >~ m)~ l ai since pN-\-q—m gives zero on the final state. Or again it may act between the potential a k and a k +\ for each k. This gives £flif II {pi+q-m)- l a t {p k +q-m)- 1 Ar=l i=Ap+l Xqipk-my^kYL (pj-m)- l aj. However, (Pk+ Q - m)- y q{p k - m)~ l = (p k -m)- l -(p k +q-m)-\ (34) so that the sum breaks into the difference of two sums, the first of which may be converted to the other by the replacement of k by k—l. There remain only the terms from the ends of the range of summation, JV-l N-l + a N II {pi-m)- x ai-a N \\ (pi+q-m)- 1 ^. t=l i=l These cancel the two terms originally discussed so that the entire effect is zero. Hence any wave emitted will satisfy dAJdx^—0. Likewise longitudinal waves (that is, waves for which A^dcp/dx^ or a=q) cannot be absorbed and will have no effect, for the matrix ele- ments for emission and absorption are similar. (We have said little more than that a potential A il =d<p/dx lt has no effect on a Dirac electron since a transformation ^' = exp( — i<p)\p removes it. It is also easy to see in coordinate representation using integrations by parts.) This has a useful practical consequence in that in computing probabilities for transition for unpolarized light one can sum the squared matrix over all four directions rather than just the two special polarization vectors. Thus suppose the matrix element for some process for light polarized in direction e M is e^M^. If the light has wave vector q^ we know from the argument above that q^M ll = 0. For unpolarized light progress- ing in the z direction we would ordinarily calculate M x 2 -\-M y 2 . But we can as well sum M x 2 -\- M y 2J r M 2 2 — M t 2 for q^Mp implies M t = M z since qt = q z for free quanta. This shows that unpolarized light is a relativistically invariant concept, and permits some simplification in computing cross sections for such light. Incidentally, the virtual quanta interact through terms like y m - • -y^hrH^k. Real processes correspond to poles in the formulae for virtual processes. The pole occurs when k 2 — 0, but it looks at first as though in the sum on all four values of n, of y m • • • y^ we would have four kinds of polarization instead of two. Now it is clear that only two perpendicular to k are effective. The usual elimination of longitudinal and scalar vir- tual photons (leading to an instantaneous Coulomb potential) can of course be performed here too (although it is not particularly useful). A typical term in a virtual transition is y ,,- • ■ y y.krH^k where the ••• represent some intervening matrices. Let us choose for the values of n, the time /, the direction of vector part K, of k f and two perpendicular directions 1, 2. We shall not change the expression for these two 1, 2 for these are represented by transverse quanta. But we must find OyrvYt)— (7k""7k)- Now k = kat— Ky K , where K= (K-K)*, and we have shown above that k replacing the 7 M gives zero. 23 Hence Ky^ is equivalent to & 4 Y< and (yr • -7<)-(7k- • ■y K ) = ((K 2 -h 2 )/K 2 )(yr • -y t ), so that on multiplying by k~ 2 d i k = d i k(k i 2 —K 2 )^ 1 the net effect is —{y t ---yt)d A k/K 2 . The y t means just scalar waves, that is, potentials produced by charge density. The fact that \/K 2 does not contain ki means that & 4 can be integrated first, resulting in an instantaneous interaction, and the d z K/K 2 is just the momentum representation of the Coulomb potential, 1/r. 9. KLEIN GORDON EQUATION The methods may be readily extended to particles of spin zero satisfying the Klein Gordon equation, 24 □V- mhlf = id (A M tA)/a^+ iA ^/dx^ -A>A rf. (35) 23 A little moi;e care is required when both 7/s act on the same particle. Define x = ^47(+^7K. an d consider (k- ■ -x) + (x- ■_■ k). Exactly this term would arise if a system, acted on by potential x carrying momentum — ft, is disturbed by an added potential ft of momentum +ft (the reversed sign of the momenta in the inter- mediate factors in the second term x- ■ k has no effect since we will later integrate over all ft). Hence as shown above the result is zero, but since (ft- • -x) + (x- • •fc) = W(7i- • -yt) — i^(7K - • % 7k) we can still conclude (7K' ■ - 7k) =k A 2 K~ 2 (yi- ■•Yt)- 24 The equations discussed in this section were deduced from the formulation of the Klein Gordon equation given in reference 5, Section 14. The function 4> in this section has only one component and is not a spinor. An alternative formal method of making the equations valid for spin zero and also for spin i is (presumably) by use of the Kemmer-Dufhn matrices /? M , satisfying the commu- tation relation If we interpret a to mean a^, rather than a M 7 M . f° r an . v a .">. a ^ of the equations in momentum space will remain formally identical to those for the spin 1/2; with the exception of those in which a denominator (p—m)~ l has been rationalized to (J>+»i)(J> 3 — m-)~ l since p 2 is no longer equal to a number, p-p. But p 3 does equal (p-p)p so that (p— m)~ l may now be interpreted as (mp+m? +p t —p'.P)(p'p—m 2 )~ 1 m~ 1 . This implies that equations _ in co- ordinate space will be valid of the function K+{2, 1) is given as A' + (2, l) = [(tV 2 +m)-»r 1 (V2 2 +02 2 )}7 + (2, 1) with v^frd/dxt,.. This is all in virtue of the fact that the many component wave function \p (5 components for spin 0, 10 for spin 1) satisfies (iV — m)\P = A\P which is formally identical to the Dirac Equation. See W. Pauli, Rev. Mod. Phys. 13, 203 (1940). 191 782 FEYNM AN The important kernel is now I+{2, 1) denned in (I, Eq. (32)). For a free particle, the wave function \p(2) satisfies +DV - m 2 $=0. At a point, 2, inside a space time region it is given by #(2)-J[*(i)az+(2,i)/a*i, -(d^/dx lli )I + (2,im il (l)dW u (as is readily shown by the usual method of demon- strating Green's theorem) the integral being over an entire 3-surface boundary of the region (with normal vector N,,). Only the positive frequency components of \f/ contribute from the surface preceding the time corre- sponding to 2, and only negative frequencies from the surface future to 2. These can be interpreted as electrons and positrons in direct analogy to the Dirac case. The right-hand side of (35) can be considered as a source of new waves and a series of terms written down to represent matrix elements for processes of increasing order. There is only one new point here, the term in AyAy by which two quanta can act at the same time. As an example, suppose three quanta or potentials, ay exp(— iq a -x), by exp(— iqb-x), and c„ exp(— iq c -x) are to act in that order on a particle of original momentum poy so that pa=po+g a and pb^pa+Qb) the final mo- mentum being p c = pb-\-g c - The matrix element is the sum of three terms (J> 2 = pypy) (illustrated in Fig. 7) (pc-c+py^iPb'-mT'ipyb+pa-b) X(pa 2 -m*)- l (pa-a+p a -a) n ,s -(pc-c+Pb-cW-mTKb-a) K ™ } -tc-b)ip a 2 -m 2 )- l (p a -a+Po-a). The first comes when each potential acts through the perturbation id(A lt \p)/dx li -\-iA ll d^/dx ll . These gradient operators in momentum space mean respectively the momentum after and before the potential Ay operates. The second term comes from by and a„ acting at the same instant and arises from the AyAy term in (a). Together by and ay carry momentum qby-\-q ai i so that after b -a operates the momentum is po-\-q a -\-Qb or p b . The final term comes from Cy and by operating together in a similar manner. The term AyAy thus permits a new type of process in which two quanta can be emitted (or absorbed, or one absorbed, one emitted) at the same time. There is no a ■ c term for the order a, b, c we have assumed. In an actual problem there would be other terms like (36) but with alterations in the order in which the quanta a, b, c act. In these terms a-c would appear. As a further example the self-energy of a particle of momentum py is (e 2 /2irim) fl(2p-k)y((p- k) 2 -m 2 )~ l X{2p-k)y-byy']d i kk- 2 Cih?), where the 5^=4 comes from the AyAy term and repre- sents the possibility of the simultaneous emission and absorption of the same virtual quantum. This integral without the C(k 2 ) diverges quadratically and would not converge if C(k 1 ) = — \ 2 /(k 2 — X 2 ). Since the interaction occurs through the gradients of the potential, we must use a stronger convergence factor, for example C(k 2 ) = X 4 (F-X 2 )- 2 , or in general (17) with f x X z G(\)d\ = 0. In this case the self-energy converges but depends quadratically on the cut-off X and is not necessarily small compared to m. The radiative corrections to scattering after mass renormalization are insensitive to the cut-off just as for the Dirac equation. When there are several particles one can obtain Bose statistics by the rule that if two processes lead to the same state but with two electrons exchanged, their amplitudes are to be added (rather than subtracted as for Fermi statistics). In this case equivalence to the second quantization treatment of Pauli and Weisskopf should be demonstrable in a way very much like that given in / (appendix) for Dirac electrons. The Bose statistics mean that the sign of contribution of a closed loop to the vacuum polarization is the opposite of what it is for the Fermi case (see I). It is (pb = p a +q) Jy, 2 irim fl(P» + pay){pb„ + p a ,){pa 2 -m 2 )- X(p b 2 -m 2 )- l -8y y (p a 2 -m 2 )- -by V {pb 2 -m 2 )-^ giving, j p = e 2 r -(qyq-8y V q 2 )\ IT L 1 X 2 1 \m 2 -q' -In— +- 6 m 2 9 3q 2 V tan0/J the notation as in (33). The imaginary part for (<7 2 )*> 2m is again positive representing the loss in the probability of finding the final state to be a vacuum, associated with the possibilities of pair production. Fermi statistics would give a gain in probability (and also a charge renormalization of opposite sign to that expected). Fig. 7. Klein-Gordon particle in three potentials, Eq. (36). The coupling to the electromagnetic field is now, for example, po-a+pa-a, and a new possibility arises, (b), of simultaneous inter- action with two quanta a -b. The propagation factor is now (/>•/»— w 2 ) -1 for a particle of momentum py. 192 QUANTUM ELECTRODYNAMICS 783 10. APPLICATION TO MESON THEORIES The theories which have been developed to describe mesons and the interaction of nucleons can be easily expressed in the language used here. Calculations, to lowest order in the interactions can be made very easily for the various theories, but agreement with experi- mental results is not obtained. Most likely all of our present formulations are quantitatively unsatisfactory. We shall content ourselves therefore with a brief sum- mary of the methods which can be used. The nucleons are usually assumed to satisfy Dirac's equation so that the factor for propagation of a nucleon of momentum p is (p—M)~ l where M is the mass of the nucleon (which implies that nucleons can be created in pairs). The nucleon is then assumed to interact with mesons, the various theories differing in the form as- sumed for this interaction. First, we consider the case of neutral mesons. The theory closest to electrodynamics is the theory of vector mesons with vector coupling. Here the factor for emis- sion or absorption of a meson is gy^ when this meson is "polarized" in the /x direction. The factor g, the "mesonic charge," replaces the electric charge e. The amplitude for propagation of a meson of momentum q in intermediate states is (q 2 — /x 2 ) -1 (rather than q~~ 2 as it is for light) where n is the mass of the meson. The neces- sary integrals are made finite by convergence factors C(q 2 — y?) as in electrodynamics. For scalar mesons with scalar coupling the only change is that one replaces the 7 M by 1 in emission and absorption. There is no longer a direction of polarization, /x, to sum upon. For pseudo- scalar mesons, pseudoscalar coupling replace 7 M by yb = iyxy y yzyt- For example, the self-energy matrix of a nucleon of momentum p in this theory is tf/irO/ 7 6 (/'-ft-M)- 1 75^(fc 2 -MVC(ft 2 -M 2 ). Other types of meson theory result from the replace- ment of 7 M by other expressions (for example by Ky^Y" - 7/yJ with a subsequent sum over all ji and v for virtual mesons). Scalar mesons with vector coupling result from the replacement of y^ by yr x q where q is the final momentum of the nucleon minus its initial mo- mentum, that is, it is the momentum of the meson if absorbed, or the negative of the momentum of a meson emitted. As is well known, this theory with neutral mesons gives zero for all processes, as is proved by our discussion on longitudinal waves in electrodynamics. Pseudoscalar mesons with pseudo-vector coupling corre- sponds to 7 M being replaced by n~ l y b q while vector mesons with tensor coupling correspond to using (2fji)~ 1 (y IJl q—qy l j). These extra gradients involve the danger of producing higher divergencies for real proc- esses. For example, y b q gives a logarithmically divergent interaction of neutron and electron. 25 Although these divergencies can be held by strong enough convergence ?5 M. Slotnick and W. Heitler, Phys. Rev. 75, 1645 (1949). factors, the results then are sensitive to the method used for convergence and the size of the cut-off values of X. For low order processes y.~ l y 5 q is equivalent to the pseudoscalar interaction 2Mpr l y b because if taken be- tween free particle wave functions of the nucleon of momenta pi and p-2 = pi~{-q, we have {ihysqui) = (H2y&(p2—pi)u\) = - (uiptynUi) - (U'ly-opiUi) = - 2M{il 2 y b uy) since 75 anticommutes with p> and pi operating on the state 2 equivalent to M as is pi on the state 1. This shows that the 75 interaction is unusually weak in the non-relativistic limit (for example the expected value of 75 for a free nucleon is zero), but since 7s 2 = 1 is not small, pseudoscalar theory gives a more important inter- action in second order than it does in first. Thus the pseudoscalar coupling constant should be chosen to fit nuclear forces including these important second order processes. 26 The equivalence of pseudoscalar and pseudo- vector coupling which holds for low order processes therefore does not hold when the pseudoscalar theory is giving its most important effects. These theories will therefore give quite different results in the majority of practical problems. In calculating the corrections to scattering of a nu- cleon by a neutral vector meson field (7^) due to the effects of virtual mesons, the situation is just as in electrodynamics, in that the result converges without need for a cut-off and depends only on gradients of the meson potential. With scalar (1) or pseudoscalar (75) neutral mesons the result diverges logarithmically and so must be cut off. The part sensitive to the cut-off, however, is directly proportional to the meson poten- tial. It may thereby be removed by a renormalization of mesonic charge g. After this renormalization the re- sults depend only on gradients of the meson potential and are essentially independent of cut-off. This is in addition to the mesonic charge renormalization coming from the production of virtual nucleon pairs by a meson, analogous to the vacuum polarization in electro- dynamics. But here there is a further difference from electrodynamics for scalar or pseudoscalar mesons in that the polarization also gives a term in the induced current proportional to the meson potential representing therefore an additional renormalization of the mass of the meson which usually depends quadratically on the cut-off. Next consider charged mesons in the absence of an electromagnetic field. One can introduce isotopic spin operators in an obvious way. (Specifically replace the neutral 75, say, by r t y b and sum over i=l, 2 where ti=t + +t_, T%=i(r+— t_) and r + changes neutron to proton (r+ on proton = 0) and r_ changes proton to neutron.) It is just as easy for practical problems simply to keep track of whether the particle is a proton or a neutron on a diagram drawn to help write down the 26 H. A. Bethe, Bull. Am. Plus. Soc. 24, 3, Z3 (Washington, 1949). 193 784 FE YN M AN matrix element. This excludes certain processes. For example in the scattering of a negative meson from qi to (72 by a neutron, the meson qz must be emitted first (in order of operators, not time) for the neutron cannot absorb the negative meson q\ until it becomes a proton. That is, in comparison to the Klein Nishina formula (15), only the analogue of second term (see Fig. 5(b)) would appear in the scattering of negative mesons by neu- trons, and only the first term (Fig. 5(a)) in the neutron scattering of positive mesons. The source of mesons of a given charge is not con- served, for a neutron capable of emitting negative me- sons may (on emitting one, say) become a proton no longer able to do so. The proof that a perturbation q gives zero, discussed for longitudinal electromagnetic waves, fails. This has the consequence that vector me- sons, if represented by the interaction y^ would not satisfy the condition that the divergence of the poten- tial is zero. The interaction is to be taken 27 as y^— m~ 2 <?m<7 in emission and as y„ in absorption if the real emission of mesons with a non-zero divergence of potential is to be avoided. (The correction term m~ 2 <7^9 gives zero in the neutral case.) The asymmetry in emission and ab- sorption is only apparent, as this is clearly the same thing as subtracting from the original y m - • -y^, a term fir 2 q- ■ q. That is, if the term —ir 2 q ll q is omitted the resulting theory describes a combination of mesons of spin one and spin zero. The spin zero mesons, coupled by vector coupling q, are removed by subtracting the term n~ 2 q- • -q. The two extra gradients q- • q make the problem of diverging integrals still more serious (for example the interaction between two protons corresponding to the exchange of two charged vector mesons depends quad- ratically on the cut-off if calculated in a straightforward way). One is tempted in this formulation to choose simply 7 M • • • 7 M and accept the admixture of spin zero mesons. But it appears that this leads in the conven- tional formalism to negative energies for the spin zero component. This shows one of the advantages of the 27 The vector meson field potentials <p„ satisfy — d/dx v {dip fl /dx„—dip v /dx l i) — n 1 ip )X ——A:TrSf i , where s^, the source for such mesons, is the matrix element of 7^ between states of neutron and proton. By taking the divergence d/dXp of both sides, conclude that d l p v /dx v = ^iry.~ i ds 1 ,/dx ll so that the original equation can be rewritten as W-<p IJL -iJ?<Pn=-'±ir{s l L+vr-d/dx ll {ds v /dx v )). The right hand side gives in momentum representation jy. — fj.~ 2 q^q v y v the left yields the (q 2 — ju 2 ) -1 and finally the interaction s^tpp in the Lagrangian gives the y M on absorption. Proceeding in this way find generally that particles of spin one can be represented by a four-vector u„. (which, for a free particle of momentum q satisfies q-u = 0). The propagation of virtual particles of momentum q from state v to n is represented by multiplication by the 4-4 matrix (or tensor) P,, v = (8^— yT^q^qv) X (? 2 — m 2 ) -1 - The first-order interaction (from the Proca equation) with an electromagnetic potential a exp( — ik-x) corresponds to multiplication by the matrix E^=(q 2 -a-\-qi-a)S liv — q2va.fi— qi^a v where q\ and _q-i = q\+k are the momenta before and after the interaction. Finally, two potentials a, b may act simultaneously, with matrix E' flv = — (a-b)du V +bna y . method of second quantization of meson fields over the present formulation. There such errors of sign are obvi- ous while here we seem to be able to write seemingly innocent expressions which can give absurd results. Pseudovector mesons with pseudovector coupling corre- spond to using 75(7^- M~ 2 (M7) for absorption and 7=,7 M for emission for both charged and neutral mesons. In the presence of an electromagnetic field, whenever the nucleon is a proton it interacts with the field in the way described for electrons. The meson interacts in the scalar or pseudoscalar case as a particle obeying the Klein-Gordon equation. It is important here to use the method of calculation of Bethe and Pauli, that is, a virtual meson is assumed to have the same "mass" dur- ing all its interactions with the electromagnetic field. The result for mass n and for (/x 2 +X 2 ) 5 are subtracted and the difference integrated over the function G(\)d\. A separate convergence factor is not provided for each meson propagation between electromagnetic interac- tions, otherwise gauge invariance is not insured. When the coupling involves a gradient, such as y^q where q is the final minus the initial momentum of the nucleon, the vector potential A must be subtracted from the momentum of the proton. That is, there is an additional coupling ±75-4 (plus when going from proton to neu- tron, minus for the reverse) representing the new possi- bility of a simultaneous emission (or absorption) of meson and photon. Emission of positive or absorption of negative virtual mesons are represented in the same term, the sign of the charge being determined by temporal relations as for electrons and positrons. Calculations are very easily carried out in this way to lowest order in g 2 for the various theories for nucleon interaction, scattering of mesons by nucleons, meson production by nuclear collisions and by gamma-rays, nuclear magnetic moments, neutron electron scattering, etc., However, no good agreement with experiment re- sults, when these are available, is obtained. Probably all of the formulations are incorrect. An uncertainty arises since the calculations are only to first order in g 2 , and are not valid if g 2 /hc is large. The author is particularly indebted to Professor H. A. Bethe for his explanation of a method of obtaining finite and gauge invariant results for the problem of vacuum polarization. He is also grateful for Professor Bethe's criticisms of the manuscript, and for innumer- able discussions during the development \>i this work. He wishes to thank Professor J. Ashkin for his careful reading of the manuscript. APPENDIX In this appendix a method will be illustrated by which the simpler integrals appearing in problems in electrodynamics can be directly .evaluated. The integrals arising in more complex processes lead to rather complicated functions, but the study of the relations of one integral to another and their expression in terms of simpler integrals may be facilitated by the methods given here. 194 QUANTUM ELECTRODYNAMICS 785 As a typical problem consider the integral (12) appearing in the first order radiationless scattering problem: fy»(Pt i)-^a(pi-k-m)- 1 y lt Jr^d*kC(k t ), (la) where we shall take C(k 2 ) to be typically -X*(**-X 8 ) -1 and d 4 k means (2Tr)~ 2 dkidk2dk i dk i . We first rationalize the factors (p-k-m)- 1 =(J>-k+m)((J>-ky-m 2 )- i obtaining, / y^-k+t^aiPi-k+f^y^d'kCik 2 ) X((/>.-ft) 2 -m 2 )- 1 ((/'2-ft) 2 -w 2 )- 1 . (2a) The matrix expression may be simplified. It appears to be best to do so after the integrations are performed. Since AB=2A B — BA where A-B = A M B M is a number commuting with all matrices, find, if R is any expression, and A a vector, since y^A= — Ay ll +2A I1 , y^ARy ll = -AypRyp+lRA. (3a) Expressions between two 7 M 's can be thereby reduced by induc- tion. Particularly useful are y lx Ay IJ ,= -2A y ll ABy ll = 2(AB+BA)- y„ABCy lx =-2CBA AA-B (4a) where A, B, C are any three vector-matrices (i.e., linear com- binations of the four 7's). In order to calculate the integral in (2a) the integral may be written as the sum of three terms (since k = k a y„), yvU>2+m)a(pi+m)y ti Ji-[y ll y a a(j) l +m)y ll + y^{p2+m)ayay^V2+ynyaay T y^Ji, (5a) where / (1 ; 8ig )=J*(i ; ;& ff ;jW r )fc-wfec(ft*) X ((fit- k)*-m*)-i((fii - ky-m*)- 1 . (6a) That is for Ji the (1; k„\ k„k T ) is replaced by 1, for Ji by k a , and for 7 3 by k a k T . More complex processes of the first order involve more factors like ((pa— k) 2 — m 2 )~ l and a corresponding increase in the number of k's which may appear in the numerator, as k„k T k v - ■ ■ . Higher order processes involving two or more virtual quanta involve similar integrals but with factors possibly involving k+k' instead of just k, and the integral extending on k- 2 d 4 kC(k 2 )k'- 2 d 4 k'C(k' 2 ). They can be simplified by methods analogous to those used on the first order integrals. The factors (p — k) 2 — m 2 may be written (p-k) 2 -tn 2 =k 2 -2p-k-A, (7a) where A = m 2 —p 2 , A l = mi i —p l 2 , etc., and we can consider dealing with cases of greater generality in that the different denominators need not have the same value of the mass m. In our specific prob- lem (6a.),pi 2 = m 2 so that Ai = 0, but we desire to work with greater generality. Now for the factor C(k 2 )/k 2 we shall use -X 2 (fc 2 -X 2 )->fc- 2 . This can be written as -X 2 /(* 2 -X 2 )ft 2 = k~ 2 C(k 2 ) = - f X2 dL(k 2 -L)- J (8a) Thus we can replace k~ 2 C(k 2 ) by (k 2 -L)~ 2 and at the end inte- grate the result with respect to L from zero to X 2 . We can for many practical purposes consider X 2 very large relative to m 2 or/) 2 . When the original integral converges even without the con- vergence factor, it will be obvious since the L integration will then be convergent to infinity. If an infra-red catastrophe exists in the integral one can simply assume quanta have a small mass X m j n and extend the integral on L from X 2 min to X 2 , rather than from zero to X s . We then have to do integrals of the form j{\; k„; k a k T )d<k(k 2 -L)^(k 2 -2p v k-A 1 )-i X(k 2 -2p 2 -k-A«)~\ (9a) where by (1; k„; k„k T ) we mean that in the place of this symbol either 1, or k a , or k„k T may stand in different cases. In more complicated problems there may be more factors (k 2 — 2pi-k — A i )~ 1 or other powers of these factors (the (k 2 — Z.) -2 may be considered as a special case of such a factor with Pi = 0, Ai = L) and further factors like k a k T k p - • ■ in the numerator. The poles in all the factors are made definite by the assumption that L, and the A's have infinitesimal negative imaginary parts. We shall do the integrals of successive complexity by induction. We start with the simplest convergent one, and show /' d<k(k 2 -L)- 3 =(8iL)- (10a) For this integral is f(2Tr)-^dk i d 3 K(k i 2 -K- K-L)~ 3 where the vector K, of magnitude AT=(K-K)* is k\, k«_, k 3 . The integral on ki shows third order poles at k i = + (K 2 +L)i and k<= -(K 2 +L)i. Imagining, in accordance with our definitions, that L has a small negative imaginary part only the first is below the real axis. The contour can be closed by an infinite semi-circle below this axis, without change of the value of the integral since the contribution from the semi-circle vanishes in the limit. Thus the contour can be shrunk about the pole k i = + (K 2 +L) i and the resulting & 4 inte- gral is — 27ri times the residue at this pole. Writing k i ={K 2 -\-L)^-\-e and expanding (k i 2 -K 2 -L)~ 3 =e- 3 (t+2(K 2 +L)i)- 3 in powers of e, the residue, being the coefficient of the term e _1 , is seen to be 6(2(K 2 + L)*)~ b so our integral is - (3i/32ir)J'~ 47rXVA'(ii?-r-Z.)- 5 ' 2 = (3/8»')(l/3I) establishing (10a). We also have J"k a d i k(k 2 — L)- 3 = from the symmetry in the k space. We write these results as (11a) (Si)j'(l; k,)d*k{k 2 -L)- 3 = (1; O)!,- 1 , where in the brackets (1; k a ) and (1; 0) corresponding entries are to be used. Substituting k= k'—p in (11a), and calling L—p 2 =A shows that (S,i)j\l; k,)d<k(k 2 -2p-k-A)- 3 =(\; p^tJt+A)-*. (12a) By differentiating both sides of (12a) with respect to A, or with respect to p T there follows directly (24i) f(U k,; k a k T )d i k(k 2 -2p-k-A)-* = -(1; p„; p„Pr-^ar(J> 2 +^))(P 2 +^)^- (13a) Further differentiations give directly successive integrals in- cluding more k factors in the numerator and higher powers of (k 2 — 2p-k — A) in the denominator. The integrals so far only contain one factor in the denominator. To obtain results for two factors we make use of the identity a- l b~ l = f* dx(ax+b(l-x))-*, (14a) (suggested by some work of Schwinger's involving Gaussian inte- grals). This represents the product of two reciprocals as a para- metric integral over one and will therefore permit integrals with two factors to be expressed in terms of one. For other powers of a, b, we make use of all of the identities, such as (15a) a -*b- l =y 2xdx(ax+b(\-x))- 3 , (14a) by successive differentiatk a integral, such as i)f(l;k a )d i k(k 2 -2p l -k-Ai)- 2 (k 2 -2p 2 -k-A 2 )- i , (16a) deducible from (14a) by successive differentiations with respect to a or b. To perform an integral, such as 195 786 k FEYNMAN write, using (15a), {k 2 -2p v k-A l )- 2 {k 2 -2p 2 -k-A 2 )^ = £ 2xdx{k 2 -2p x -k-A x )^, where px = xPi+(l-x)p 2 and A z = xA 1 -\-(l-x)A 2 , (17a) (note that A z is not equal to m 2 — pi 1 ) so that the expression (16a) is (8i)f i 2xdxf(l;k«)d i k(k 2 -2p z -k-A x )- :i which may now be evaluated by (12a) and is (16a) = £ {\;p xa )2xdx(p x 2 +A z )~\ (18a) where p x , A x are given in (17a). The integral in (18a) is elementary, being the integral of ratio of polynomials, the denominator of second degree in x. The general expression although readily ob- tained is a rather complicated combination of roots and logarithms. Other integrals can be obtained again by parametric differentia- tion. For example differentiation of (16a), (18a) with respect to A 2 or p2 T gives (8i)f(l;k a ; k < ,k T )d i k(k 2 -2p r k-Ai)- i (k i -2p2-k-A2)- 2 X2x(l-x)dx(J> x *+A x )-*, (19a) again leading to elementary integrals. As an example, consider the case that the second factor is just (k 2 -L)^ and in the first put Pi=p, A 1 = A. Then p x = xp, A z = xA+(l — x)L. There results (Si)f(l;k a ; k a k T )d i k{k 2 -L)- 2 {,k 2 -2p-k-A)^ = -£ (liXP^xtprPr-farixtp+At)) X2x(l-x)dx(x*p*+A x )- i . (20a) Integrals with three factors can be reduced to those involving two by using (14a) again. They, therefore, lead to integrals with two parameters (e.g., see application to radiative correction to scattering below). The methods of calculation given in this paper are deceptively simple when applied to the lower order processes. For processes of increasingly higher orders the complexity and difficulty in- creases rapidly, and these methods soon become impractical in their present form. A. Self-Energy The self-energy integral (19) is (e'/TrDfy^ip-k-m^y^d^kCik 2 ), (19) so that it requires that we find (using the principle of (8a)) the integral on L from to X 2 of fy li (p-k+m)y ll d , k(k 2 -L)- 2 (k 2 -2p-k)- 1 , since {p-k) 2 -m 2 =k 2 -2p-k, SLSp 2 =m 2 . This is of the form (16a) with Ai = L, />i = 0, A 2 = 0, p2=p so that (18a) gives, since p x =(l-x)p, A z = xL, (8») J*(l ; k <7 )d*k(k 2 -L)^(k 2 -2p ■ k)~ l = £ (l;(l-x)p a )2xdx((l-x)W+xL)-\ or performing the integral on L, as in (8), (8i)/(l; K)d^kk^C{k 2 ){k 2 -2p-k)^ Jo ,r " (l-x) 2 m 2 Assuming now that \ 2 »m 2 we neglect (1— x)hn 2 relative to xX 2 in the argument of the logarithm, which then becomes (X 2 /m 2 )(x/(l-x) 2 ). Then since fHx ln(*(l -*)"*) = 1 and f a l (l-x)dx ln(x(l-x)- 2 )= -(1/4) find (8i)f(l;k <r )k- i C(k i )d l k(k 2 -2p-kr i so that substitution into (19) (after the (p-k-m)' 1 in (19) is replaced by (p—k-\-m)(k 2 — 2p-k)~ 1 ) gives (19) = ( e 2 /8^)T,[^+w)(21n(X 2 A« 2 ) + 2) -/»(ln(X 2 /m 2 )-i)] 7M = (e 2 /8 1 r)[8w(ln(X 2 /« 2 )+l)-/>(2 1n(X 2 /w 2 ) + 5)], (20) using (4a) to remove the 7^'s. This agrees with Eq. (20) of the text, and gives the self-energy (21) when p is replaced by m. B. Corrections to Scattering The term (12) in the radiationless scattering, after rationalizing the matrix denominators and using pi 2 = p 2 2 = m 2 requires the integrals (9a), as we have discussed. This is an integral with three denominators which we do in two stages. First the factors (k 2 — 2pi-k) and (k 2 — 2p 2 -k) are combined by a parameter y; {k 2 -2p v k)-\k 2 -2p 2 -k)^ = £ dy(k 2 -2p y -k)^, from (14a) where Pu=ypi+i\~y)p2- (2ia) We therefore need the integrals (M)/(l; V; k„k T )d i k{k 2 -L)~ 2 (k 2 -2p y -k)- Ji , (22a) which we will then integrate with respect to y from to 1. Next we do the integrals (22a) immediately from (20a) with p =p y , A = : (22a) = -£ £ (1; xp y ,; x 2 p ya p yT -hd, T (x 2 p y 2 +(l-x)L))2x(l-x)dx(x 2 p y 2 +L(l-x))- 2 dy. We now turn to the integrals on L as required in (8a). The first term, (1), in (1; k„; k a k T ) gives no trouble for large L, but if L is put equal to zero there results x~~ 2 p y ~ 2 which leads to a diverging integral on x as x—*0. This infra-red catastrophe is analyzed by using Xmin 2 for the lower limit of the L integral. For the last term the upper limit of L must be kept as X 2 . Assuming X ro , n 2 <3C/> y 2 <3C X* the x integrals which remain are trivial, as in the self-energy case. One finds - (U)J{k 2 - X min 2 )-W *kC(k 2 - X min 2 ) (k 2 - 2p x ■ kT\k 2 - 2p2 ■ k)~ l = £p y -*dy\n(p y 2 /\ miD 2 ) (23a) -(8i)fk a k- 2 d i kC(k 2 )(k 2 -2p l -k)- 1 (k 2 -2p2-k)-i = l£ P».P*-*dy, (24a) -(8i)fk a k T k-WkC(k 2 )(k 2 -2p 1 -k)-Kk 2 -2p2-kr i = £ PvPvTpv-Vy-farf* dy InOtyT 8 ) + * 5 "- (25a) The integrals on y give, £ p y ~*dy IntaAmin- 2 ) = 4(m 2 sin20)-i[~0 l n (wX mi »- 1 ) — f ata.nada\, (26a) £ Pv<rPv~ 2 dy = 8(m i sin20)- 1 (J>i ff +£2„), (27a) £ PycP V rPy- i dy = e(2m 2 An29)-^{pi 9 +p lT )(pu+ptr) +«^V?T(l-0ctn0), (28a) £ dy ln(X 2 A,- 2 ) = ln(X7w 2 )+2(l-0ctn0). (29a) 196 QUANTUM ELECTRODYNAMICS 787 These integrals on y were performed as follows. Since p2=p\+q where q is the momentum carried by the potential, it follows from p 2 2=p l t=m? that 2pi-q=—q 2 so that since py=pi+q(l — y), p u 2 = m 2 —q 2 y(l — y). The substitution 2y— l = tana:/tan0 where 6 is defined by 4»z 2 sin 2 = <7 2 is useful for it means p v 2 = m 2 sec 2 a/sec 2 d and p v ~Hy= (m 2 sin20) _1 da! where a. goes from —6 to -f 0. These results are substituted into the original scattering formula (2a), giving (22). It has been simplified by frequent use of the fact that pi operating on the initial state is m, and likewise pi when it appears at the left is replacable by m. (Thus, to simplify: y^ptapiy li = -Ipidpt by (4a), = -2(p 2 -q)a(J) l +q) = -2(m-q)a(m+q). A term like qaq= — q 2 a-\-2{a-q)q is equivalent to just — q 2 a since q=p 2 —pi = m — m has zero matrix element.) The renormalization term requires the corresponding integrals for the special case 9 = 0. C. Vacuum Polarization The expressions (32) and (32') for / M „ in the vacuum polariza- tion problem require the calculation of the integral /„,(>»-) = --. fsply ll (p-^q^m)y v ip+iq+m)2d i p X{{p-hqT~-m 2 )-KiP+\qY-ni 2 r\ (32) where we have replaced p by p—\q to simplify the calculation somewhat. We shall indicate the method of calculation by studying the integral, /( IW 2)= J p.p T d*p(U>-lq)*-m*)-K(P+iqV The factors in the denominator, p 2 — p ■ q— m 2 -\-\q 2 and p 2J rp-q — m 2 -\-\q 2 are combined as usual by (8a) but for symmetry we substitute .v=j(1 + ij), (1— x) = i(l — rf) and integrate jj from -1 to +1: I{m 2 ) =f2 p < ,p r d>p(fi 2 -r ) p-q-m 2 +\q 2 )- 2 dr l j2. (30a) But the integral on p will not be found in our list for it is badly divergent. However, as discussed in Section 7, Eq. (32') we do not wish I(m 2 ) but rather /o™[/(m 2 )-/(w 2 +X 2 )]G(X)JX. We can calculate the difference I{m 2 ) — I(m 2 -\-\ 2 ) by first calculating the derivative I'(m 2 +L) of / with respect to m 2 at m 2 -\-L and later integrating L from zero to X 2 . By differentiating (30a), with respect to m 2 find, I'(m 2 +L)= f_* p a p r d*p{p 2 --op-q- m 2 - L+\q 2 THr,. This still diverges, but we can differentiate again to get I"(m 2 +L)=?>f* l i p G p T d i p{p 2 -r l p-q-m 2 -L+\q 2 )- i dr 1 -(8i) -1 XT (.h%qrD-*-$5,, T iri)dv (31a) (where Z? = j(tj 2 — \)q 2 +m 2 -\-L), which now converges and has been evaluated by (13a) with p=\riq and A — m 2 +L— \q 2 . Now to get /' we may integrate /" with respect to L as an indefinite integral and we may choose any convenient arbitrary constant. This is because a constant C in /' will mean a term — CX 2 in I(m 2 ) — 7(w 2 +X 2 ) which vanishes since we will integrate the results times G(\)d\ and Jo'°\ 2 G(\)d\ = 0. This means that the logarithm appearing on integrating L in (31a) presents no problem. We may take I'(m 2 +L) = (Si)" 1 /* 1 lWq,qrD- l +i8, T \nDy n +C&. T , a subsequent integral on L and finally on 77 presents no new problems. There results - mfp.Prd*pi(p- \q) 2 ~ m*)-K(P+\q)*- m 2 )~ l p 4m 2 - q 2 / B \, , \ 2 1 + 5„r[(X 2 + w 2 )ln(X 2 ,«- 2 + l)-C'X 2 ], (32a) where we assume X 2 ^>w 2 and have put some terms into the arbi- trary constant C which is independent of X 2 (but in principle could depend on q 2 ) and which drops out in the integral on G{\)d\. We have set q 2 = 4w 2 sin 2 0. In a very similar way the integral with m 2 in the numerator can be worked out. It is, of course, necessary to differentiate this m 2 also when calculating /' and /". There results -(8i)fm 2 d<p((p-iq) 2 -m 2 ri((p+lq) 2 - m 2 )-i = 4m 2 (l-e>ctn9)-g 2 /3+2(X 2 +w 2 )ln(X 2 m- 2 + l)-C"X 2 ), (33a) with another unimportant constant C". The complete problem re- quires the further integral, -(.8i)f(UP*)d i P((p-lq) 2 --m*)-K(P+$q) i -mT 1 = (l,O)(4(l-0ctn0) + 2 1n(X 2 w- 2 )). (34a) The value of the integral (34a) times m 2 differs from (33a), of course, because the results on the right are not actually the inte- grals on the left, but rather equal their actual value minus their value for ;w 2 =»z 2 +X 2 . Combining these quantities, as required by (32), dropping the constants C, C" and evaluating the spur gives (33). The spurs are evaluated in the usual way, noting that the spur of any odd number of 7 matrices vanishes and Sp(AB)=Sp(BA) for arbi- trary A, B. The 57>(1)=4 and we also have iSpl(pi+m l ){p2-m2)l= pi- pt-mum, (35a) \Spl{p l +m l ){p2-m 2 )iJ> 3 +m i )(p i -m i )2 = (pr p2—mim2)(p3- pi-m 3 m t ) — (Pi-p3-mim 3 )(p2-pi—m 2 m i ) + (.pi-pi—mmt)(p2-pa—tnim 3 ), (36a) where pi, nu are arbitrary four-vectors and constants. It is interesting that the terms of order X 2 lnX 2 go out, so that the charge renormalization depends only logarithmically on X 2 . This is not true for some of the meson theories. Electrodynamics is suspiciously unique in the mildness of its divergence. D. More Complex Problems Matrix elements for complex problems can be set up in a manner analogous to that used for the simpler cases. We give three illustrations; higher order corrections to the Miller scatter- Fig. 8. The interaction between two electrons t< One adds the contribution of every figure involvi quanta, Appendix D. 197 788 FEYNMAN ing, to the Compton scattering, and the interaction of a neutron with an electromagnetic field. For the M011er scattering, consider two electrons, one in state Mi of momentum pi and the other in state uz of momentum pi. Later they are found in states z< 3 , Pz and u±, p t . This may happen (first order in e 2 /hc) because they exchange a quantum of momen- tum q=Pi—pi=pi—p2 in the manner of Eq. (4) and Fig. 1. The matrix element for this process is proportional to (translating (4) to momentum space) («47m«2)(«»Ym m i)? -2 - (37a) We shall discuss corrections to (37a) to the next order in e 2 /hc. (There is also the possibility that it is the electron at 2 which finally arrives at 3, the electron at 1 going to 4 through the ex- change of quantum of momentum p 3 —p2. The amplitude for this process, (ufY l iUi)(ii 3 y ll U2)(j>3—p2)~ 2 , must be subtracted from (37a) in accordance with the exclusion principle. A similar situa- tion exists to each order so that we need consider in detail only the corrections to (37a), reserving to the last the subtraction of the same terms with 3, 4 exchanged.) One reason that (37a) is modified is that two quanta may be exchanged, in the manner of Fig. 8a. The total matrix element for all exchanges of this type is (e?/wi)J (uiyvipi- k- m)- 1 y lt ui)(u i y v (j)2+ k— m)-^^) ■k-'iq-kyWk, (38a) as is clear from the figure and the general rule that electrons of momentum p contribute in amplitude (p—mY 1 between inter- actions t m , and that quanta of momentum k contribute ft -2 . In integrating on d*k and summing over fi and v. we add all alterna- tives of the type of Fig. 8a. If the time of absorption, y^, of the quantum k by electron 2 is later than the absorption, y„, of q— k, this corresponds to the virtual state pz+k being a positron (so that (38a) contains over thirty terms of the conventional method of analysis). In integrating over all these alternatives we have considered all possible distortions of Fig. 8a which preserve the order of events along the trajectories. We have not included the possibilities corresponding to Fig. 8b, however. Their contribution is means that one adds with equal weight the integrals corresponding to each topologically distinct figure. To this same order there are also the possibilities of Fig. 8d which give (*/*i)f{u*Yp(Pi- k-m)-^^) Xiua^+q-k-my^udk-Kq-krWk, (39a) as is readily verified by labeling the diagram. The contributions of all possible ways that an event can occur are to be added. This w* I) J (uwvipz—k—m) 1 y ll .{pi — h—m) l y v Ui) This integral on k will be seen to be precisely the integral (12) for the radiative corrections to scattering, which we have worked out. The term may be combined with the renormalization terms result- ing from the difference of the effects of mass change and the terms, Figs. 8f and 8g. Figures 8e, 8h, and 8i are similarly analyzed. Finally the term Fig. 8c is clearly related to our vacuum polarization problem, and when integrated gives a term propor- tional to (u4y»U2)(u i y v Ui)J llv q~ i . If the charge is renormalized the term ln(X/ra) in J ^ in (33) is omitted so there is no remaining dependence on the cut-off. The only new integrals we require are the convergent integrals (38a) and (39a). They can be simplified by rationalizing the de- nominators and combining them by (14a). For example (38a) in- volves the factors (k 2 -2p v k)~ 1 {k 2 +2p2- k)- 1 k~ 2 (q 2 + k 2 -2q- k)~ 2 . The first two may be combined by (14a) with a parameter x, and the second pair by an expression obtained by differentiation (15a) with respect to b and calling the parameter y. There results a factor (k 2 -2p x - k)~ 2 (k 2 +yq 2 —2yq- k)' 4 so that the integrals on d 4 k now involve two factors and can be performed by the methods given earlier in the appendix. The subsequent integrals on the parameters x and y are complicated and have not been worked out in detail. Working with charged mesons there is often a considerable re- duction of the number of terms. For example, for the interaction between protons resulting from the exchange of two mesons only the term corresponding to Fig. 8b remains. Term 8a, for example, is impossible, for if the first proton emits a positive meson the second cannot absorb it directly for only neutrons can absorb positive mesons. As a second example, consider the radiative correction to the Compton scattering. As seen from Eq. (15) and Fig. 5 this scatter- ing is represented by two terms, so that we can consider the cor- rections to each one separately. Figure 9 shows the types of terms arising from corrections to the term of Fig. 5a. Calling k the momentum of the virtual quantum, Fig. 9a gives an integral fy„(p2- k- ?n)- 1 e 2 (pi+q i - k-m^e^-k- m^y^lrWk, convergent without cut-off and reducible by the methods outlined in this appendix. The other terms are relatively easy to evaluate. Terms b and c of Fig. 9 are closely related to radiative corrections (although somewhat more difficult to evaluate, for one of the states is not that of a free electron, (J>i-\-q) 2 ^m 2 ). Terms e, f are renormaliza- tion terms. From term d must be subtracted explicitly the effect of mass Am, as analyzed in Eqs. (26) and (27) leading to (28) with p'=pi+q, a = e2, b = e\. Terms g, h give zero since the vacuum polarization has zero effect on free light quanta, q\ 2 = Q, q2 2 = 0. The total is insensitive to the cut-off X. The result shows an infra-red catastrophe, the largest part of the effect. When cut-off at X m j n , the effect proportional to ln(m/X m i n ) goes as (e 2 /w) ln(m/X min )(l-20 ctn20), (40a) times the uncorrected amplitude, where (/> 2 — />0 2 = 4;n 2 sin 2 0. This is the same as for the radiative correction to scattering for a deflection pi—p\. This is physically clear since the long wave quanta are not effected by short-lived intermediate states. The infra-red effects arise 28 from a final adjustment of the field from the asymptotic coulomb field characteristic of the electron of Fig. 9. Radiative correction to the Compton scattering term (a) of Fig. 5. Appendix D. F. Bloch and A. Nordsieck, Phys. Rev. 52, 54 (1937). 198 mm^ QUANTUM ELECTRODYNAMICS 789 momentum pi before the collision to that characteristic of an electron moving in a new direction pi after the collision. The complete expression for the correction is a very complicated expression involving transcendental integrals. As a final example we consider the interaction of a neutron with an electromagnetic field in virtue of the fact that the neutron may emit a virtual negative meson. We choose the example of pseudo- scalar mesons with pseudovector coupling. The change in ampli- tude due to an electromagnetic field A = aexp(—iq-x) determines the scattering of a neutron by such a field. In the limit of small q it will vary as qa—aq which represents the interaction of a par- ticle possessing a magnetic moment. The first-order interaction between an electron and a neutron is given by the same calculation by considering the exchange of a quantum between the electron and the nucleon. In this case a M is q~ 2 times the matrix element of 7 M between the initial and final states of the electron, the states differing in momentum by q. The interaction may occur because the neutron of momentum pi emits a negative meson becoming a proton which proton inter- acts with the field and then reabsorbs the meson (Fig. 10a). The matrix for this process is (J>2=pi+q), f(y s k)(p i -k-M)- 1 a(Pi-k-M)-Kysk)(k?-v?)-Wk. (41a) Alternatively it may be the meson which interacts with the field. We assume that it does this in the manner of a scalar potential satisfying the Klein Gordon Eq. (35), (Fig. 10b) ■f(yM(J>i-ki-M)-Ky>ki)(W-S)-i Xih-a+ki-a)^ TWki, (42a) where we have put ki=ki+q. The change in sign arises because the virtual meson is negative. Finally there are two terms arising from the y b a part of the pseudovector coupling (Figs. 10c, lOd) J'iyMiPi-k-Myiiy^iW-SrWk, (43a) and fiy^iPi-k-Mr^yskKV-SrWk. (44a) Using convergence factors in the manner discussed in the section on meson theories each integral can be evaluated and the results combined. Expanded in powers of q the first term gives the mag- netic moment of the neutron and is insensitive to the cut-off, the next gives the scattering amplitude of slow electrons on neutrons, and depends logarithmically on the cut-off. The expressions may be simplified and combined somewhat before integration. This makes the integrals a little easier and also shows the relation to the case of pseudoscalar coupling. For example in (41a) the final ysk can be written as ys(k—pi+M) since pi = M when operating on the initial neutron state. This is ^5*2 p-k P. c. Fig. 10. According to the meson theory a neutron interacts with an electromagnetic potential a by first emitting a virtual charged meson. The figure illustrates the case for a pseudoscalar meson with pseudovector coupling. Appendix D. (Pi— k— M)y b +2Mys since 75 anticommutes with pi and k. The first term cancels the (pi—k—M) -1 and gives a term which just cancels (43a). In a like manner the leading factor y b k in (41a) is written as — 2Myi— 7s(/>2— k— M), the second term leading to a simpler term containing no (pi—k — M)' 1 factor and combining with a similar one from (44a). One simplifies the 7 5 fci and 75^2 in (42a) in an analogous way. There finally results terms like (41a), (42a) but with pseudoscalar coupling 2M75 instead of 75ft, no terms like (43a) or (44a) and a remainder, representing the difference in effects of pseudovector and pseudoscalar coupling. The pseudoscalar terms do not depend sensitively on the cut-off, but the difference term depends on it logarithmically. The differ- ence term affects the electron-neutron interaction but not the magnetic moment of the neutron. Interaction of a proton with an electromagnetic potential can be similarly analyzed. There is an effect of virtual mesons on the electromagnetic properties of the proton even in the case that the mesons are neutral. It is analogous to the radiative corrections to the scattering of electrons due to virtual photons. The sum of the magnetic moments of neutron and proton for charged mesons is the same as the proton moment calculated for the corresponding neutral mesons. In fact it is readily seen by comparing diagrams, that for arbitrary q, the scattering matrix to first order in the electromagnetic potential for a proton according to neutral meson theory is equal, if the mesons were charged, to the sum of the matrix for a neutron and the matrix for a proton. This is true, for any type or mixtures of meson coupling, to all orders in the coupling (neglecting the mass difference of neutron and proton). MARSTON SCIENCE LIBRARY MARSTON SCIENCE LIBRARY Date Due Due Returned Due Returned raw 05 1999 ' ; ■ ' " \ /I/ JA* f ? 2(lflfi/ UNIVERSITY OF FLORIDA 3 1262 05971 7719 * HECKMAN BINDERY INC. APR 98 Bound - To -Pleasl 5 N.MANCHESTER INDIANA 46962 '